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How to Use Graphs to Solve Equation Systems: Word Problems

In the mathematical universe, graphs and equation systems converge in harmony, guiding us to elegant solutions for complex problems. The merger of these mathematical concepts provides a robust method for tackling a variety of word problems. This article will elucidate a step-by-step approach to utilizing graphs to solve equation systems in word problems.

A Step-by-step Guide to Using Graphs to Solve Equation Systems: Word Problems

Equation systems represent a collection of two or more equations with a similar set of variables. Graphs, on the other hand, visualize equations by representing them in the Cartesian plane. This combination presents a compelling method to interpret and solve a multitude of real-life situations, neatly encapsulated in word problems.

Step 1: Understanding the Word Problem

Every word problem narrates a scenario requiring a mathematical solution. Here, you must identify the unknown variables and frame equations representing the given situation. For instance, let’s consider a problem relating two different taxi services with their fare structures given. The unknowns here could be the distance covered or the total fare, giving us two linear equations.

Step 2: Formulating the Equation System

The next step involves translating the word problem into an equation system . The equations should correspond to the relationships or conditions described in the problem. With our taxi services example, we might generate equations like \(y=3x+2\) and \(y=4x+1\) where \(y\) represents the total fare, \(x\) the distance and the constants are based on the fare structures given.

Step 3: Graphing the Equations

With the equation system in place, the next course of action is to draw the graph for each equation . You plot each equation on the same Cartesian plane. This visual representation effectively demonstrates the relationship between the variables in a dynamic way that numbers alone cannot.

Step 4: Identifying the Solution

The point where the two graphs intersect is the solution to the system. This intersection signifies the value of the variables that satisfy both equations simultaneously. In our taxi services scenario, this point represents the distance at which both services charge the same fare.

Step 5: Verifying the Solution

A key step in mathematics is always to verify your solution . Plug the obtained solution back into the original equations to ensure they hold true. If they do, you’ve found the correct solution. If not, revisit your calculations.

by: Effortless Math Team about 12 months ago (category: Articles )

Effortless Math Team

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1.20: Word Problems for Linear Equations

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Word problems are important applications of linear equations. We start with examples of translating an English sentence or phrase into an algebraic expression.

Example 18.1

Translate the phrase into an algebraic expression:

a) Twice a variable is added to 4

Solution: We call the variable \(x .\) Twice the variable is \(2 x .\) Adding \(2 x\) to 4 gives:

\[4 + 2x\nonumber\]

b) Three times a number is subtracted from 7.

Solution: Three times a number is \(3 x .\) We need to subtract \(3 x\) from 7. This means:\

\[7-3 x\nonumber\]

c) 8 less than a number.

Solution: The number is denoted by \(x .8\) less than \(x\) mean, that we need to subtract 8 from it. We get:

\[x-8\nonumber\]

For example, 8 less than 10 is \(10-8=2\).

d) Subtract \(5 p^{2}-7 p+2\) from \(3 p^{2}+4 p\) and simplify.

Solution: We need to calculate \(3 p^{2}+4 p\) minus \(5 p^{2}-7 p+2:\)

\[\left(3 p^{2}+4 p\right)-\left(5 p^{2}-7 p+2\right)\nonumber\]

Simplifying this expression gives:

\[\left(3 p^{2}+4 p\right)-\left(5 p^{2}-7 p+2\right)=3 p^{2}+4 p-5 p^{2}+7 p-2 =-2 p^{2}+11 p-2\nonumber\]

e) The amount of money given by \(x\) dimes and \(y\) quarters.

Solution: Each dime is worth 10 cents, so that this gives a total of \(10 x\) cents. Each quarter is worth 25 cents, so that this gives a total of \(25 y\) cents. Adding the two amounts gives a total of

\[10 x+25 y \text{ cents or } .10x + .25y \text{ dollars}\nonumber\]

Now we deal with word problems that directly describe an equation involving one variable, which we can then solve.

Example 18.2

Solve the following word problems:

a) Five times an unknown number is equal to 60. Find the number.

Solution: We translate the problem to algebra:

\[5x = 60\nonumber\]

We solve this for \(x\) :

\[x=\frac{60}{5}=12\nonumber\]

b) If 5 is subtracted from twice an unknown number, the difference is \(13 .\) Find the number.

Solution: Translating the problem into an algebraic equation gives:

\[2x − 5 = 13\nonumber\]

We solve this for \(x\). First, add 5 to both sides.

\[2x = 13 + 5, \text{ so that } 2x = 18\nonumber\]

Dividing by 2 gives \(x=\frac{18}{2}=9\).

c) A number subtracted from 9 is equal to 2 times the number. Find the number.

Solution: We translate the problem to algebra.

\[9 − x = 2x\nonumber\]

We solve this as follows. First, add \(x\) :

\[9 = 2x + x \text{ so that } 9 = 3x\nonumber\]

Then the answer is \(x=\frac{9}{3}=3\)

d) Multiply an unknown number by five is equal to adding twelve to the unknown number. Find the number.

Solution: We have the equation:

\[5x = x + 12.\nonumber\]

Subtracting \(x\) gives

\[4x = 12.\nonumber\]

Dividing both sides by 4 gives the answer: \(x=3\).

e) Adding nine to a number gives the same result as subtracting seven from three times the number. Find the number.

Solution: Adding 9 to a number is written as \(x+9,\) while subtracting 7 from three times the number is written as \(3 x-7\). We therefore get the equation:

\[x + 9 = 3x − 7.\nonumber\]

We solve for \(x\) by adding 7 on both sides of the equation:

\[x + 16 = 3x.\nonumber\]

Then we subtract \(x:\)

\[16 = 2x.\nonumber\]

After dividing by \(2,\) we obtain the answer \(x=8\)

The following word problems consider real world applications. They require to model a given situation in the form of an equation.

Example 18.3

a) Due to inflation, the price of a loaf of bread has increased by \(5 \%\). How much does the loaf of bread cost now, when its price was \(\$ 2.40\) last year?

Solution: We calculate the price increase as \(5 \% \cdot \$ 2.40 .\) We have

\[5 \% \cdot 2.40=0.05 \cdot 2.40=0.1200=0.12\nonumber\]

We must add the price increase to the old price.

\[2.40+0.12=2.52\nonumber\]

The new price is therefore \(\$ 2.52\).

b) To complete a job, three workers get paid at a rate of \(\$ 12\) per hour. If the total pay for the job was \(\$ 180,\) then how many hours did the three workers spend on the job?

Solution: We denote the number of hours by \(x\). Then the total price is calculated as the price per hour \((\$ 12)\) times the number of workers times the number of hours \((3) .\) We obtain the equation

\[12 \cdot 3 \cdot x=180\nonumber\]

Simplifying this yields

\[36 x=180\nonumber\]

Dividing by 36 gives

\[x=\frac{180}{36}=5\nonumber\]

Therefore, the three workers needed 5 hours for the job.

c) A farmer cuts a 300 foot fence into two pieces of different sizes. The longer piece should be four times as long as the shorter piece. How long are the two pieces?

\[x+4 x=300\nonumber\]

Combining the like terms on the left, we get

\[5 x=300\nonumber\]

Dividing by 5, we obtain that

\[x=\frac{300}{5}=60\nonumber\]

Therefore, the shorter piece has a length of 60 feet, while the longer piece has four times this length, that is \(4 \times 60\) feet \(=240\) feet.

d) If 4 blocks weigh 28 ounces, how many blocks weigh 70 ounces?

Solution: We denote the weight of a block by \(x .\) If 4 blocks weigh \(28,\) then a block weighs \(x=\frac{28}{4}=7\)

How many blocks weigh \(70 ?\) Well, we only need to find \(\frac{70}{7}=10 .\) So, the answer is \(10 .\)

Note You can solve this problem by setting up and solving the fractional equation \(\frac{28}{4}=\frac{70}{x}\). Solving such equations is addressed in chapter 24.

e) If a rectangle has a length that is three more than twice the width and the perimeter is 20 in, what are the dimensions of the rectangle?

Solution: We denote the width by \(x\). Then the length is \(2 x+3\). The perimeter is 20 in on one hand and \(2(\)length\()+2(\)width\()\) on the other. So we have

\[20=2 x+2(2 x+3)\nonumber\]

Distributing and collecting like terms give

\[20=6 x+6\nonumber\]

Subtracting 6 from both sides of the equation and then dividing both sides of the resulting equation by 6 gives:

\[20-6=6 x \Longrightarrow 14=6 x \Longrightarrow x=\frac{14}{6} \text { in }=\frac{7}{3} \text { in }=2 \frac{1}{3} \text { in. }\nonumber\]

f) If a circle has circumference 4in, what is its radius?

Solution: We know that \(C=2 \pi r\) where \(C\) is the circumference and \(r\) is the radius. So in this case

\[4=2 \pi r\nonumber\]

Dividing both sides by \(2 \pi\) gives

\[r=\frac{4}{2 \pi}=\frac{2}{\pi} \text { in } \approx 0.63 \mathrm{in}\nonumber\]

g) The perimeter of an equilateral triangle is 60 meters. How long is each side?

Solution: Let \(x\) equal the side of the triangle. Then the perimeter is, on the one hand, \(60,\) and on other hand \(3 x .\) So \(3 x=60\) and dividing both sides of the equation by 3 gives \(x=20\) meters.

h) If a gardener has \(\$ 600\) to spend on a fence which costs \(\$ 10\) per linear foot and the area to be fenced in is rectangular and should be twice as long as it is wide, what are the dimensions of the largest fenced in area?

Solution: The perimeter of a rectangle is \(P=2 L+2 W\). Let \(x\) be the width of the rectangle. Then the length is \(2 x .\) The perimeter is \(P=2(2 x)+2 x=6 x\). The largest perimeter is \(\$ 600 /(\$ 10 / f t)=60\) ft. So \(60=6 x\) and dividing both sides by 6 gives \(x=60 / 6=10\). So the dimensions are 10 feet by 20 feet.

i) A trapezoid has an area of 20.2 square inches with one base measuring 3.2 in and the height of 4 in. Find the length of the other base.

Solution: Let \(b\) be the length of the unknown base. The area of the trapezoid is on the one hand 20.2 square inches. On the other hand it is \(\frac{1}{2}(3.2+b) \cdot 4=\) \(6.4+2 b .\) So

\[20.2=6.4+2 b\nonumber\]

Multiplying both sides by 10 gives

\[202=64+20 b\nonumber\]

Subtracting 64 from both sides gives

\[b=\frac{138}{20}=\frac{69}{10}=6.9 \text { in }\nonumber\]

and dividing by 20 gives

Exit Problem

Write an equation and solve: A car uses 12 gallons of gas to travel 100 miles. How many gallons would be needed to travel 450 miles?

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System-of-Equations Word Problems

Exercises More Exercises

Very commonly, system-of-equations word problems involves mixtures or combinations of some sort. For instance:

A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totalled $487 . The second order was for 6 bushes and 2 trees, and totalled $232 . The bills do not list the per-item price. What were the costs of one bush and of one tree?

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System of Equations Word Problems

I could try to add the bushes and trees, to get 19 bushes and 6 trees, but this wouldn't get me anywhere, because I don't have subtotals for the bushes and trees. So I'll pick variables:

number of bushes: B

number of trees: t

With these variables, I can set up a system of equations; each equation will represent one of the transactions they've given me:

1st order: 13B + 4 t = 487

2nd order: 6B + 2 t = 232

Multiplying the second row by –2 , I get:

13B + 4 t = 487

–12B – 4 t = –464

Adding down the t -terms cancel out, leaving me with B = 23 . Back-solving, I get that t  = 47 . Of course, the exercise didn't ask for the values of the two variables. Translating back into English, my solution is:

bushes: $23 each

trees: $47 each

You probably remember the "distance" word problems where you had a boat going with the current and then against the current, or a plane going with the wind (that is, having a tailwind) and then against the wind (that is, having a headwind). Once you learn how to solve systems of equations, you'll see more of these sorts of exercises.

A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed?

When they ask me about the speed "in still air" (for planes) or "in still water" (for boats), they are referring to the speedometer reading; they are referring only to the powered input, irrespective of outside influences.

On some very windy day, you can watch birds flapping frantically in the air, trying to cross the street, say, from the tree in your front yard. No matter how hard they flapped, they made little or no forward progress; sometimes, a bird will even appear to fly backwards! Did this mean that the bird wasn't actually flapping? No; it meant that the bird's attempted speed (how fast the flapping would have moved the bird on a windless day) was not fast enough to usefully counteract the wind hitting it in the face. The bird's "speed in still air", less the wind's speed in the opposite direction, was close to zero, or even negative.

The same concept applies to machinery. If a boat's motor is chugging away at 10 miles an hour (according to the speedometer), but the boat is facing a water current of 15 miles an hour in the opposite direction, then the boat will end up going backwards at five miles an hour. In other words, the speedometer reading is not always the actual speed.

Returning to the exercise:

I'll pick variables and set up a system. In this case, I'll use:

plane's speedometer: p

windspeed: w

When the plane is going "with" the wind, the plane's powered speed and the windspeed will add together; when the plane is going "against" the wind, the windspeed will be subtracted from the plane's speedometer reading (that is, from the engines' actual output).

In each case, the "distance" equation will be "(the combined speed) times (the time spent at that speed) equals (the total distance travelled)":

with the jetstream: ( p + w )(3) = 1800

against the jetstream: ( p – w )(4) = 1800

Rather than multiply through, I notice that, if I divide off the 3 and the 4, I'll have a system that's already set for solving by addition :

p + w = 600

p – w = 450

Then, by adding down, I get:

Back-solving, I see that the windspeed w must be 75 mph.

jet's speed: 525 mph

windspeed: 75 mph

Another topic you might see (if not now, then later in calculus) is decomposing rational expressions using partial fractions .

Find the partial fraction decomposition of the following:

The denominator of the polynomial fraction they've given me factors as:

( x + 2)( x + 1)( x – 1)

These factors will be the denominators in the partial-fraction decomposition. That is, I'll be looking for the values of A , B , and C which will complete the following:

The above expression is meant to be equal to the original fraction they gave me. Setting them equal and then multiplying both sides by the common denominator, I get:

5 x + 7 = A( x + 1)( x – 1) + B( x + 2)( x – 1) + C( x + 2)( x + 1)

= A( x 2 – 1) + B( x 2 + x – 2) + C( x 2 + 3 x + 2)

= (A + B + C) x 2 + (B + 3C) x + (–A – 2B + 2C)1

The standard way of solving this big messy equation is the process of "comparing coefficients". Two polynomials are equal only if the coefficients of their terms are equal. This is why I grouped my terms the way I did in the last line above; I've grouped everything that was multiplied by x 2 , everything that was multiplied by x , and everything that was multiplied by 1 (that is, everything that was just a constant, with no variable part).

On the left-hand side, I've got 5 x  + 7 , which has no term with an x 2 , so I'll need to think of " 5 x  + 7 " as " 0 x 2  + 5 x  + 7 ". This will allow me to create new equations, based on the fact that the coefficients on either side of the "equals" sign have to be the same. This gives me:

x 2 : A + B + C = 0

x : B + 3C = 5

1: –A – 2B + 2C = 7

Solving this system, I get A = –1 , B = –1 , and C = 2 . Then the partial fraction decomposition is:

Whatever you do, don't panic when you face a systems-of-equations word problem. If you take them step-by-step, they're usually pretty do-able. That said, it would probably be to your benefit if you did extra practice problems, just to help you get in the swing of things. With any luck, your tests will then go a little faster.

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\(\textbf{1)}\) Joe and Steve are saving money. Joe starts with $105 and saves $5 per week. Steve starts with $5 and saves $15 per week. After how many weeks do they have the same amount of money? Show Equations \(y= 5x+105,\,\,\,y=15x+5\) Show Answer 10 weeks ($155)

\(\textbf{2)}\) mike and sarah collect rocks. together they collected 50 rocks. mike collected 10 more rocks than sarah. how many rocks did each of them collect show equations \(m+s=50,\,\,\,m=s+10\) show answer mike collected 30 rocks, sarah collected 20 rocks., \(\textbf{3)}\) in a classroom the ratio of boys to girls is 2:3. there are 25 students in the class. how many are girls show equations \(b+g=50,\,\,\,3b=2g\) show answer 15 girls (10 boys), \(\textbf{4)}\) kyle makes sandals at home. the sandal making tools cost $100 and he spends $10 on materials for each sandal. he sells each sandal for $30. how many sandals does he have to sell to break even show equations \(c=10x+100,\,\,\,r=30x\) show answer 5 sandals ($150), \(\textbf{5)}\) molly is throwing a beach party. she still needs to buy beach towels and beach balls. towels are $3 each and beachballs are $4 each. she bought 10 items in total and it cost $34. how many beach balls did she get show equations show answer 4 beachballs (6 towels), \(\textbf{6)}\) anna volunteers at a pet shelter. they have cats and dogs. there are 36 pets in total at the shelter, and the ratio of dogs to cats is 4:5. how many cats are at the shelter show equations \(c+d=40,\,\,\,5d=4c\) show answer 20 cats (16 dogs), \(\textbf{7)}\) a store sells oranges and apples. oranges cost $1.00 each and apples cost $2.00 each. in the first sale of the day, 15 fruits were sold in total, and the price was $25. how many of each type of frust was sold show equations \(o+a=15,\,\,\,1o+2a=25\) show answer 10 apples and 5 oranges, \(\textbf{8)}\) the ratio of red marbles to green marbles is 2:7. there are 36 marbles in total. how many are red show equations \(r+g=36,\,\,\,7r=2g\) show answer 8 red marbles (28 green marbles), \(\textbf{9)}\) a tennis club charges $100 to join the club and $10 for every hour using the courts. write an equation to express the cost \(c\) in terms of \(h\) hours playing tennis. show equation the equation is \(c=10h+100\), \(\textbf{10)}\) emma and liam are saving money. emma starts with $80 and saves $10 per week. liam starts with $120 and saves $6 per week. after how many weeks will they have the same amount of money show equations \(e = 10x + 80,\,\,\,l = 6x + 120\) show answer 10 weeks ($180 each), \(\textbf{11)}\) mark and lisa collect stamps. together they collected 200 stamps. mark collected 40 more stamps than lisa. how many stamps did each of them collect show equations \(m + l = 200,\,\,\,m = l + 40\) show answer mark collected 120 stamps, lisa collected 80 stamps., \(\textbf{12)}\) in a classroom, the ratio of boys to girls is 3:5. there are 40 students in the class. how many are boys show equations \(b + g = 40,\,\,\,5b = 3g\) show answer 15 boys (25 girls), \(\textbf{13)}\) lisa is selling handmade jewelry. the materials cost $60, and she sells each piece for $20. how many pieces does she have to sell to break even show equations \(c=60,\,\,\,r=20x\) show answer 3 pieces, \(\textbf{14)}\) tom is buying books and notebooks for school. books cost $15 each, and notebooks cost $3 each. he bought 12 items in total, and it cost $120. how many notebooks did he buy show equations \(b + n = 12,\,\,\,15b+3n=120\) show answer 5 notebooks (7 books), \(\textbf{15)}\) emily volunteers at an animal shelter. they have rabbits and guinea pigs. there are 36 animals in total at the shelter, and the ratio of guinea pigs to rabbits is 4:5. how many guinea pigs are at the shelter show equations \(r + g = 36,\,\,\,5g=4r\) show answer 16 guinea pigs (20 rabbits), \(\textbf{16)}\) mike and sarah are going to a theme park. mike’s ticket costs $40, and sarah’s ticket costs $30. they also bought $20 worth of food. how much did they spend in total show equations \(m + s + f = t,\,\,\,m=40,\,\,\,s=30,\,\,\,f=20\) show answer they spent $90 in total., \(\textbf{17)}\) the ratio of red marbles to blue marbles is 2:3. there are 50 marbles in total. how many are blue show equations \(r + b = 50,\,\,\,3r=2b\) show answer 30 blue marbles (20 red marbles), \(\textbf{18)}\) a pizza restaurant charges $12 for a large pizza and $8 for a small pizza. if a customer buys 5 pizzas in total, and it costs $52, how many large pizzas did they buy show equations \(l + s = 5,\,\,\,12l+8s=52\) show answer they bought 3 large pizzas (2 small pizzas)., \(\textbf{19)}\) the area of a rectangle is 48 square meters. if the length is 8 meters, what is the width of the rectangle show equations \(a=l\times w,\,\,\,l=8,\,\,\,a=48\) show answer the width is 6 meters., \(\textbf{20)}\) two numbers have a sum of 50. one number is 10 more than the other. what are the two numbers show equations \(x+y=50,\,\,\,x=y+10\) show answer the numbers are 30 and 20., \(\textbf{21)}\) a store sells jeans for $40 each and t-shirts for $20 each. in the first sale of the day, they sold 8 items in total, and the price was $260. how many of each type of item was sold show equations \(j+t=8,\,\,\,40j+20t=260\) show answer 5 jeans and 3 t-shirts were sold., \(\textbf{22)}\) the ratio of apples to carrots is 3:4. there are 28 fruits in total. how many are apples show equations \(\)a+c=28,\,\,\,4a=3c show answer there are 12 apples and 16 carrots., \(\textbf{23)}\) a phone plan costs $30 per month, and there is an additional charge of $0.10 per minute for calls. write an equation to express the cost \(c\) in terms of \(m\) minutes. show equation the equation is \(\)c=30+0.10m, \(\textbf{24)}\) a triangle has a base of 8 inches and a height of 6 inches. calculate its area. show equations \(a=0.5\times b\times h,\,\,\,b=8,\,\,\,h=6\) show answer the area is 24 square inches., \(\textbf{25)}\) a store sells shirts for $25 each and pants for $45 each. in the first sale of the day, 4 items were sold, and the price was $180. how many of each type of item was sold show equations \(t+p=4,\,\,\,25t+45p=180\) show answer 0 shirts and 4 pants were sold., \(\textbf{26)}\) a garden has a length of 12 feet and a width of 10 feet. calculate its area. show equations \(a=l\times w,\,\,\,l=12,\,\,\,w=10\) show answer the area is 120 square feet., \(\textbf{27)}\) the sum of two consecutive odd numbers is 56. what are the two numbers show equations \(x+y=56,\,\,\,x=y+2\) show answer the numbers are 27 and 29., \(\textbf{28)}\) a toy store sells action figures for $15 each and toy cars for $5 each. in the first sale of the day, 10 items were sold, and the price was $110. how many of each type of item was sold show equations \(a+c=10,\,\,\,15a+5c=110\) show answer 6 action figures and 4 toy cars were sold., \(\textbf{29)}\) a bakery sells pie for $2 each and cookies for $1 each. in the first sale of the day, 14 items were sold, and the price was $25. how many of each type of item was sold show equations \(p+c=14,\,\,\,2p+c=25\) show answer 11 pies and 3 cookies were sold., \(\textbf{for 30-33}\) two car rental companies charge the following values for x miles. car rental a: \(y=3x+150 \,\,\) car rental b: \(y=4x+100\), \(\textbf{30)}\) which rental company has a higher initial fee show answer company a has a higher initial fee, \(\textbf{31)}\) which rental company has a higher mileage fee show answer company b has a higher mileage fee, \(\textbf{32)}\) for how many driven miles is the cost of the two companies the same show answer the companies cost the same if you drive 50 miles., \(\textbf{33)}\) what does the \(3\) mean in the equation for company a show answer for company a, the cost increases by $3 per mile driven., \(\textbf{34)}\) what does the \(100\) mean in the equation for company b show answer for company b, the initial cost (0 miles driven) is $100., \(\textbf{for 35-39}\) andy is going to go for a drive. the formula below tells how many gallons of gas he has in his car after m miles. \(g=12-\frac{m}{18}\), \(\textbf{35)}\) what does the \(12\) in the equation represent show answer andy has \(12\) gallons in his car when he starts his drive., \(\textbf{36)}\) what does the \(18\) in the equation represent show answer it takes \(18\) miles to use up \(1\) gallon of gas., \(\textbf{37)}\) how many miles until he runs out of gas show answer the answer is \(216\) miles, \(\textbf{38)}\) how many gallons of gas does he have after 90 miles show answer the answer is \(7\) gallons, \(\textbf{39)}\) when he has \(3\) gallons remaining, how far has he driven show answer the answer is \(162\) miles, \(\textbf{for 40-42}\) joe sells paintings. each month he makes no commission on the first $5,000 he sells but then makes a 10% commission on the rest., \(\textbf{40)}\) find the equation of how much money x joe needs to sell to earn y dollars per month. show answer the answer is \(y=.1(x-5,000)\), \(\textbf{41)}\) how much does joe need to sell to earn $10,000 in a month. show answer the answer is \($105,000\), \(\textbf{42)}\) how much does joe earn if he sells $45,000 in a month show answer the answer is \($4,000\), see related pages\(\), \(\bullet\text{ word problems- linear equations}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- averages}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- consecutive integers}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- distance, rate and time}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- break even}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- ratios}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- age}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- mixtures and concentration}\) \(\,\,\,\,\,\,\,\,\), linear equations are a type of equation that has a linear relationship between two variables, and they can often be used to solve word problems. in order to solve a word problem involving a linear equation, you will need to identify the variables in the problem and determine the relationship between them. this usually involves setting up an equation (or equations) using the given information and then solving for the unknown variables . linear equations are commonly used in real-life situations to model and analyze relationships between different quantities. for example, you might use a linear equation to model the relationship between the cost of a product and the number of units sold, or the relationship between the distance traveled and the time it takes to travel that distance. linear equations are typically covered in a high school algebra class. these types of problems can be challenging for students who are new to algebra, but they are an important foundation for more advanced math concepts. one common mistake that students make when solving word problems involving linear equations is failing to set up the problem correctly. it’s important to carefully read the problem and identify all of the relevant information, as well as any given equations or formulas that you might need to use. other related topics involving linear equations include graphing and solving systems. understanding linear equations is also useful for applications in fields such as economics, engineering, and physics., about andymath.com, andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. if you have any requests for additional content, please contact andy at [email protected] . he will promptly add the content. topics cover elementary math , middle school , algebra , geometry , algebra 2/pre-calculus/trig , calculus and probability/statistics . in the future, i hope to add physics and linear algebra content. visit me on youtube , tiktok , instagram and facebook . andymath content has a unique approach to presenting mathematics. the clear explanations, strong visuals mixed with dry humor regularly get millions of views. we are open to collaborations of all types, please contact andy at [email protected] for all enquiries. to offer financial support, visit my patreon page. let’s help students understand the math way of thinking thank you for visiting. how exciting.

Solve a system of equations by graphing: word problems

Avery wants to buy a charm bracelet. Vindale Fine Jewelry charges $12 per charm, plus $30 for the bracelet. Hardin Jewelers, in contrast, charges $16 per charm and $10 for the bracelet. If Avery wants to add a certain number of charms to her bracelet, the cost will be the same at either jewelry shop. What would the total cost of the bracelet be?

1 Expert Answer

Philip P. answered • 01/31/22

Effective and Patient Math Tutor

Let x be the number of charms and y be the cost of the bracelet.

Cost of Vindale bracelet: y = 12x + 30

Cost of Hardin bracelet: y = 16x + 10

Graph the two linear equations. The point where they intersect is where they are equal and the y-value of the intersection is the cost.

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Solve Word Problems involving Graphs

a. The beetle’s number of wing vibrations is the same as the difference between the fly and honeybee’s.

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Graphing Word Problem Worksheets

How Do Graphs Help Us Explain Data? Graphs are one of the common ways of representing data. They help us visually understand numbers and figures. Graphs are mainly used for representing a relationship between two identities or variables. Any form of graph will give you a visual display of data that was either given to you or that you actually collected. Being able to see the data in this form allows you to understand trends in single data sets. When you compare multiple graphs, you can see how one trial or event compares to a previous event. This can tell us which event was more or less effective. It can also allow us to predict the outcome of future events or trials, with a high level of certainty. There is tremendous need for people in the workforce that can create, read, and interpret graphs to make predictions. Atmospheric scientists study and interpret graphs of temperature, pressure, and precipitation to make solid guesses on what the weather has in store for us. Medical researchers track the progress of clinical trials to provide us with medicine that can save our lives or at least enhance it. Actuaries track risk data that helps us understand risks in certain industries. Economists track changes in society and business data to make predictions on the future of the world's economy. This series of worksheets and lessons shows students how to tackle word problems that include the use of graphs.

Aligned Standard: 3.MD.B.3 and 5.OA.4

• Arnie's Ice Cream Sales Lesson - We look at the sales over all four seasons. Winter is rough for this shop.
• Superhero Guided Lesson - Time to look at the worldwide movie box office sales of some of your favorite Superhero movies.
• Line Chart Guided Lesson Explanation - Use the graph to infer how much each movie grossed.
• Pizza Sales Bar Graph 1 - Track the sales from two classes frozen pizza fundraiser.
• Pie Graph - Class Hobbies Sheet 2 - I would have thought video games and sports would have been higher.
• Count and Create Bar Graph Worksheet 3 - Make your own bar graph from the data that is given.
• Pie Chart Word Problems 3-Pack - Let's break down those circle charts in high detail. Members only access.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

We work with fundraisers and double line graphs in this series.

• Homework 1 - Blackbourne Middle School had 5 different fundraisers during the school year to raise funds for a big end of year party. After each fundraiser was completed, the data was entered into this bar graph. Use the graph to answer the following questions.
• Homework 2 - Ben and Brianna have been collecting cards for a popular trading card game for 6 years. After the first year, they decided to compare how many each of them had at the end of every year. The graph below is the record of those 5 years.
• Homework 3 - The graph below represents the amount of lemonade sold per hour during the day. Use the line graph to answer the questions.

Practice Worksheets

We look at all the major forms of interpreting graphs.

• Practice 1 - Mrs. Smith's class was asked what their favorite fruit was. The graph shows the number of students who liked each fruit. Use the graph to answer the questions
• Practice 2 - Was there an increase or decrease in the amount of yards Tony mowed between Tuesday and Wednesday?
• Practice 3 - Zoe worked all summer to earn money for a new laptop. She kept track of her earnings from her separated jobs to see which job she earned more from.

Math Skill Quizzes

This quiz set builds through repetition and skill level.

• Quiz 1 - How many people live in Frankfort, Topeka and Helena combined?
• Quiz 2 - Beth, and Aiden are working to finish reading their novels for English class. They decided to see who could read the most in one week. Below, the line graph is a recording of how many pages they each read during one week.
• Quiz 3 - A local carnival took a survey to find out which of their rides teenagers liked the most. Their results are recorded on the bar graph below.

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Course: Algebra 1   >   Unit 7

• Writing two-variable inequalities word problem
• Solving two-variable inequalities word problem
• Interpreting two-variable inequalities word problem
• Two-variable inequalities word problems
• Modeling with systems of inequalities
• Writing systems of inequalities word problem
• Solving systems of inequalities word problem

Graphs of systems of inequalities word problem

• Systems of inequalities word problems
• Graphs of two-variable inequalities word problem
• Inequalities (systems & graphs): FAQ
• Creativity break: What can we do to expand our creative skills?

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