solving linear equations variables on both sides assignment

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

8.5: Solve Equations with Variables and Constants on Both Sides (Part 2)

• Last updated
• Save as PDF
• Page ID 6910

Solve Equations Using a General Strategy

Each of the first few sections of this chapter has dealt with solving one specific form of a linear equation. It’s time now to lay out an overall strategy that can be used to solve any linear equation. We call this the general strategy. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.

HOW TO: USE A GENERAL STRATEGY FOR SOLVING LINEAR EQUATIONS

Step 1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.

Step 2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.

Step 3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.

Step 4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation.

Step 5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

Example \(\PageIndex{11}\):

Solve: 3(x + 2) = 18.

Exercise \(\PageIndex{21}\):

Solve: 5(x + 3) = 35.

Exercise \(\PageIndex{22}\):

Solve: 6(y − 4) = −18.

Example \(\PageIndex{12}\):

Solve: −(x + 5) = 7.

Exercise \(\PageIndex{23}\):

Solve: −(y + 8) = −2.

Exercise \(\PageIndex{24}\):

Solve: −(z + 4) = −12.

Example \(\PageIndex{13}\):

Solve: 4(x − 2) + 5 = −3.

Exercise \(\PageIndex{25}\):

Solve: 2(a − 4) + 3 = −1.

Exercise \(\PageIndex{26}\):

Solve: 7(n − 3) − 8 = −15.

Example \(\PageIndex{14}\):

Solve: 8 − 2(3y + 5) = 0.

Be careful when distributing the negative.

Exercise \(\PageIndex{27}\):

Solve: 12 − 3(4j + 3) = −17.

\(j = \frac{5}{3}\)

Exercise \(\PageIndex{28}\):

Solve: −6 − 8(k − 2) = −10.

\(k = \frac{5}{2}\)

Example \(\PageIndex{15}\):

Solve: 3(x − 2) − 5 = 4(2x + 1) + 5.

Exercise \(\PageIndex{29}\):

Solve: 6(p − 3) − 7 = 5(4p + 3) − 12.

Exercise \(\PageIndex{30}\):

Solve: 8(q + 1) − 5 = 3(2q − 4) − 1.

Example \(\PageIndex{16}\):

Solve: \(\dfrac{1}{2}\)(6x − 2) = 5 − x.

Exercise \(\PageIndex{31}\):

Solve: \(\dfrac{1}{3}\)(6u + 3) = 7 − u.

Exercise \(\PageIndex{32}\):

Solve: \(\dfrac{2}{3}\)(9x − 12) = 8 + 2x.

In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

Example \(\PageIndex{17}\):

Solve: 0.24(100x + 5) = 0.4(30x + 15).

Exercise \(\PageIndex{33}\):

Solve: 0.55(100n + 8) = 0.6(85n + 14).

Exercise \(\PageIndex{34}\):

Solve: 0.15(40m − 120) = 0.5(60m + 12).

ACCESS ADDITIONAL ONLINE RESOURCES

Solving Multi-Step Equations

Solve an Equation with Variable Terms on Both Sides

Solving Multi-Step Equations (L5.4)

Solve an Equation with Variables and Parentheses on Both Sides

Practice Makes Perfect

Solve an equation with constants on both sides.

In the following exercises, solve the equation for the variable.

• 6x − 2 = 40
• 7x − 8 = 34
• 11w + 6 = 93
• 14y + 7 = 91
• 3a + 8 = −46
• 4m + 9 = −23
• −50 = 7n − 1
• −47 = 6b + 1
• 25 = −9y + 7
• 29 = −8x − 3
• −12p − 3 = 15
• −14q − 15 = 13

Solve an Equation with Variables on Both Sides

• 8z = 7z − 7
• 9k = 8k − 11
• 4x + 36 = 10x
• 6x + 27 = 9x
• c = −3c − 20
• b = −4b − 15
• 5q = 44 − 6q
• 7z = 39 − 6z
• 3y + \(\dfrac{1}{2}\) = 2y
• 8x + \(\dfrac{3}{4}\) = 7x
• −12a − 8 = −16a
• −15r − 8 = −11r

Solve an Equation with Variables and Constants on Both Sides

In the following exercises, solve the equations for the variable.

• 6x − 15 = 5x + 3
• 4x − 17 = 3x + 2
• 26 + 8d = 9d + 11
• 21 + 6 f = 7 f + 14
• 3p − 1 = 5p − 33
• 8q − 5 = 5q − 20
• 4a + 5 = − a − 40
• 9c + 7 = −2c − 37
• 8y − 30 = −2y + 30
• 12x − 17 = −3x + 13
• 2z − 4 = 23 − z
• 3y − 4 = 12 − y
• \(\dfrac{5}{4}\)c − 3 = \(\dfrac{1}{4}\)c − 16
• \(\dfrac{4}{3}\)m − 7 = \(\dfrac{1}{3}\)m − 13
• 8 − \(\dfrac{2}{5}\)q = \(\dfrac{3}{5}\)q + 6
• 11 − \(\dfrac{1}{4}\)a = \(\dfrac{3}{4}\)a + 4
• \(\dfrac{4}{3}\)n + 9 = \(\dfrac{1}{3}\)n − 9
• \(\dfrac{5}{4}\)a + 15 = \(\dfrac{3}{4}\)a − 5
• \(\dfrac{1}{4}\)y + 7 = \(\dfrac{3}{4}\)y − 3
• \(\dfrac{3}{5}\)p + 2 = \(\dfrac{4}{5}\)p − 1
• 14n + 8.25 = 9n + 19.60
• 13z + 6.45 = 8z + 23.75
• 2.4w − 100 = 0.8w + 28
• 2.7w − 80 = 1.2w + 10
• 5.6r + 13.1 = 3.5r + 57.2
• 6.6x − 18.9 = 3.4x + 54.7

Solve an Equation Using the General Strategy

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

• 5(x + 3) = 75
• 4(y + 7) = 64
• 8 = 4(x − 3)
• 9 = 3(x − 3)
• 20(y − 8) = −60
• 14(y − 6) = −42
• −4(2n + 1) = 16
• −7(3n + 4) = 14
• 3(10 + 5r) = 0
• 8(3 + 3p) = 0
• \(\dfrac{2}{3}\)(9c − 3) = 22
• \(\dfrac{3}{5}\)(10x − 5) = 27
• 5(1.2u − 4.8) = −12
• 4(2.5v − 0.6) = 7.6
• 0.2(30n + 50) = 28
• 0.5(16m + 34) = −15
• −(w − 6) = 24
• −(t − 8) = 17
• 9(3a + 5) + 9 = 54
• 8(6b − 7) + 23 = 63
• 10 + 3(z + 4) = 19
• 13 + 2(m − 4) = 17
• 7 + 5(4 − q) = 12
• −9 + 6(5 − k) = 12
• 15 − (3r + 8) = 28
• 18 − (9r + 7) = −16
• 11 − 4(y − 8) = 43
• 18 − 2(y − 3) = 32
• 9(p − 1) = 6(2p − 1)
• 3(4n − 1) − 2 = 8n + 3
• 9(2m − 3) − 8 = 4m + 7
• 5(x − 4) − 4x = 14
• 8(x − 4) − 7x = 14
• 5 + 6(3s − 5) = −3 + 2(8s − 1)
• −12 + 8(x − 5) = −4 + 3(5x − 2)
• 4(x − 1) − 8 = 6(3x − 2) − 7
• 7(2x − 5) = 8(4x − 1) − 9

Everyday Math

• Making a fence Jovani has a fence around the rectangular garden in his backyard. The perimeter of the fence is 150 feet. The length is 15 feet more than the width. Find the width, w, by solving the equation 150 = 2(w + 15) + 2w.
• Concert tickets At a school concert, the total value of tickets sold was $1,506. Student tickets sold for $6 and adult tickets sold for $9. The number of adult tickets sold was 5 less than 3 times the number of student tickets. Find the number of student tickets sold, s, by solving the equation 6s + 9(3s − 5) = 1506.
• Coins Rhonda has $1.90 in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the number of nickels, n, by solving the equation 0.05n + 0.10(2n − 1) = 1.90.
• Fencing Micah has 74 feet of fencing to make a rectangular dog pen in his yard. He wants the length to be 25 feet more than the width. Find the length, L, by solving the equation 2L + 2(L − 25) = 74.

Writing Exercises

203. When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient as the variable side? 204. Solve the equation 10x + 14 = −2x + 38, explaining all the steps of your solution. 205. What is the first step you take when solving the equation 3 − 7(y − 4) = 38? Explain why this is your first step. 206. Solve the equation 1 4 (8x + 20) = 3x − 4 explaining all the steps of your solution as in the examples in this section. 207. Using your own words, list the steps in the General Strategy for Solving Linear Equations. 208. Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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] ."