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8.E: Solving Linear Equations (Exercises)

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8.1 - Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, determine whether the given number is a solution to the equation.

• x + 16 = 31, x = 15
• w − 8 = 5, w = 3
• −9n = 45, n = 54
• 4a = 72, a = 18

In the following exercises, solve the equation using the Subtraction Property of Equality.

• y + 2 = −6
• a + \(\dfrac{1}{3} = \dfrac{5}{3}\)
• n + 3.6 = 5.1

In the following exercises, solve the equation using the Addition Property of Equality.

• u − 7 = 10
• x − 9 = −4
• c − \(\dfrac{3}{11} = \dfrac{9}{11}\)
• p − 4.8 = 14

In the following exercises, solve the equation.

• n − 12 = 32
• y + 16 = −9
• f + \(\dfrac{2}{3}\) = 4
• d − 3.9 = 8.2
• y + 8 − 15 = −3
• 7x + 10 − 6x + 3 = 5
• 6(n − 1) − 5n = −14
• 8(3p + 5) − 23(p − 1) = 35

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

• The sum of −6 and m is 25.
• Four less than n is 13.

In the following exercises, translate into an algebraic equation and solve.

• Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?
• Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?
• Peter paid $9.75 to go to the movies, which was $46.25 less than he paid to go to a concert. How much did he pay for the concert?
• Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week?

8.2 - Solve Equations using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division Property of Equality.

• 13a = −65
• 0.25p = 5.25
• −y = 4

In the following exercises, solve each equation using the Multiplication Property of Equality.

• \(\dfrac{n}{6}\) = 18
• y −10 = 30
• 36 = \(\dfrac{3}{4}\)x
• \(\dfrac{5}{8} u = \dfrac{15}{16}\)

In the following exercises, solve each equation.

• −18m = −72
• \(\dfrac{c}{9}\) = 36
• 0.45x = 6.75
• \(\dfrac{11}{12} = \dfrac{2}{3} y\)
• 5r − 3r + 9r = 35 − 2
• 24x + 8x − 11x = −7−14

8.3 - Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the equations with constants on both sides.

• 8p + 7 = 47
• 10w − 5 = 65
• 3x + 19 = −47
• 32 = −4 − 9n

In the following exercises, solve the equations with variables on both sides.

• 7y = 6y − 13
• 5a + 21 = 2a
• k = −6k − 35
• 4x − \(\dfrac{3}{8}\) = 3x

In the following exercises, solve the equations with constants and variables on both sides.

• 12x − 9 = 3x + 45
• 5n − 20 = −7n − 80
• 4u + 16 = −19 − u
• \(\dfrac{5}{8} c\) − 4 = \(\dfrac{3}{8} c\) + 4

In the following exercises, solve each linear equation using the general strategy.

• 6(x + 6) = 24
• 9(2p − 5) = 72
• −(s + 4) = 18
• 8 + 3(n − 9) = 17
• 23 − 3(y − 7) = 8
• \(\dfrac{1}{3}\)(6m + 21) = m − 7
• 8(r − 2) = 6(r + 10)
• 5 + 7(2 − 5x) = 2(9x + 1) − (13x − 57)
• 4(3.5y + 0.25) = 365
• 0.25(q − 8) = 0.1(q + 7)

8.4 - Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation by clearing the fractions.

• \(\dfrac{2}{5} n − \dfrac{1}{10} = \dfrac{7}{10}\)
• \(\dfrac{1}{3} x + \dfrac{1}{5} x = 8\)
• \(\dfrac{3}{4} a − \dfrac{1}{3} = \dfrac{1}{2} a + \dfrac{5}{6}\)
• \(\dfrac{1}{2}\)(k + 3) = \(\dfrac{1}{3}\)(k + 16)

In the following exercises, solve each equation by clearing the decimals.

• 0.8x − 0.3 = 0.7x + 0.2
• 0.36u + 2.55 = 0.41u + 6.8
• 0.6p − 1.9 = 0.78p + 1.7
• 0.10d + 0.05(d − 4) = 2.05

PRACTICE TEST

• \(\dfrac{23}{5}\)
• n − 18 = 31
• 4y − 8 = 16
• −8x − 15 + 9x − 1 = −21
• −15a = 120
• \(\dfrac{2}{3}\)x = 6
• x + 3.8 = 8.2
• 10y = −5y + 60
• 8n + 2 = 6n + 12
• 9m − 2 − 4m + m = 42 − 8
• −5(2x + 1) = 45
• −(d + 9) = 23
• 2(6x + 5) − 8 = −22
• 8(3a + 5) − 7(4a − 3) = 20 − 3a
• \(\dfrac{1}{4} p + \dfrac{1}{3} = \dfrac{1}{2}\)
• 0.1d + 0.25(d + 8) = 4.1
• Translate and solve: The difference of twice x and 4 is 16.
• Samuel paid $25.82 for gas this week, which was $3.47 less than he paid last week. How much did he pay last week?

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/[email protected] ."

Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

• For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
• For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

• Add or Subtract the same value from both sides
• Clear out any fractions by Multiplying every term by the bottom parts
• Divide every term by the same nonzero value
• Combine Like Terms
• Expanding (the opposite of factoring) may also help
• Recognizing a pattern, such as the difference of squares
• Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

• solve Quadratic Equations
• solve Radical Equations
• solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

• Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
• Show all the steps , so it can be checked later (by you or someone else)

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