2.7.1: Solving Linera Equations

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Learning Objectives

By the end of this section, you will be able to:

Solve equations using a general strategy

It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will.

Beginning by simplifying each side of the equation makes the remaining steps easier.

Exercise $\PageIndex{1}$: How to Solve Linear Equations Using the General Strategy

Solve: $-6(x + 3) = 24$.

Answer: $This figure is a table that has three columns and five rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Simplify each side of the equation as much as possible.” The text in the second cell reads: “Use the Distributive Property. Notice that each side of the equation is simplified as much as possible.” The third cell contains the equation negative 6 times x plus 3, where x plus 3 is in parentheses, equals 24. Below this is the same equation with the negative 6 distributed across the parentheses: negative 6x minus 18 equals 24.$ $Step 2. Collect all variable terms on one side of the equation. Here there's nothing more to do, since there is only one x on the left side.$ $In the third row of the table, the first cell says: “Step 3. Collect constant terms on the other side of the equation. In the second cell, the instructions say: “To get constants only on the right, add 18 to each side. Simplify.” The third cell contains the same equation with 18 added to both sides: negative 6x minus 18 plus 18 equals 24 plus 18. Below this is the equation negative 6x equals 42.$ $Step 4. Make the coefficient of x one. Here we divide both sides by -6 and simplify!$ $In the fifth row of the table, the first cell says: “Step 5. Check the solution.” In the second cell, the instructions say: “Let x equal negative 7. Simplify. Multiply.” In the third cell, there is the instruction: “Check,” and to the right of this is the original equation again: negative 6 times x plus 3, with x plus 3 in parentheses, equal 24. Below this is the same equation with negative 7 substituted in for x: negative 6 times negative 7 plus 3, with negative 7 plus 3 in parentheses, might equal 24. Below this is the equation negative 6 times negative 4 might equal 24. Below this is the equation 24 equals 24, with a check mark next to it.$

Exercise $\PageIndex{2}$

Solve: $5(x + 3)=35$

Answer: $x = 4$

Exercise $\PageIndex{3}$

Solve: $6(y - 4) = -18$

Answer: $y = 1$

GENERAL STRATEGY FOR SOLVING LINEAR EQUATIONS.

Simplify each side of the equation as much as possible.
Use the Distributive Property to remove any parentheses.
Combine like terms.
Collect all the variable terms on one side of the equation.
Use the Addition or Subtraction Property of Equality.
Collect all the constant terms on the other side of the equation.
Use the Addition or Subtraction Property of Equality.
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.
Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

Exercise $\PageIndex{4}$

Solve: $-(y + 9) = 8$

Answer

		$.$
Simplify each side of the equation as much as possible by distributing.		$.$
The only y term is on the left side, so all variable terms are on the left side of the equation.
Add 9 to both sides to get all constant terms on the right side of the equation.		$.$
Simplify.		$.$
Rewrite −y as −1y.		$.$
Make the coefficient of the variable term to equal to 1 by dividing both sides by −1.		$.$
Simplify.		$.$
Check:	$.$
Let y=−17.	$.$
	$.$
	$.$

Exercise $\PageIndex{5}$

Solve: $-(y + 8) = -2$

Answer: $y = -6$

Exercise $\PageIndex{6}$

Solve: $-(z + 4) = -12$

Answer: $z = 8$

Exercise $\PageIndex{7}$

Solve: $5(a - 3) + 5 = -10$

Answer

		$.$
Simplify each side of the equation as much as possible.
Distribute.		$.$
Combine like terms.		$.$
The only a term is on the left side, so all variable terms are on one side of the equation.
Add 10 to both sides to get all constant terms on the other side of the equation.		$.$
Simplify.		$.$
Make the coefficient of the variable term to equal to 11 by dividing both sides by 55.		$.$
Simplify.		$.$
Check:	$.$
Let a=0.	$.$
	$.$
	$.$
	$.$

Exercise $\PageIndex{8}$

Solve: $2(m - 4) + 3 = -1$

Answer: $m = 2$

Exercise $\PageIndex{9}$

Solve:$7(n - 3) - 8 = -15$

Answer: $n = 2$

Exercise $\PageIndex{10}$

Solve: $\frac{2}{3}(6m - 3) = 8 - m$

Answer

		$.$
Distribute.		$.$
Add m to get the variables only to the left.		$.$
Simplify.		$.$
Add 2 to get constants only on the right.		$.$
Simplify.		$.$
Divide by 5.		$.$
Simplify.		$.$
Check:	$.$
Let m=2.	$.$
	$.$
	$.$
	$.$

Exercise $\PageIndex{11}$

Solve: $\frac{1}{3}(6u + 3) = 7 - u$

Answer: $u = 2$

Exercise $\PageIndex{12}$

Solve: $\frac{2}{3}(9x - 12) = 8 + 2x$

Answer: $x = 4$

Exercise $\PageIndex{13}$

Solve: $8 - 2(3y + 5) = 0$

Answer

	$.$
Simplify—use the Distributive Property.	$.$
Combine like terms.	$.$
Add 2 to both sides to collect constants on the right.	$.$
Simplify.	$.$
Divide both sides by −6−6.	$.$
Simplify.	$.$
Check: Let y=−13. $.$

Exercise $\PageIndex{14}$

Solve: $12 - 3(4j + 3) = -17$

Answer: $j = \frac{5}{3}$

Exercise $\PageIndex{15}$

Solve: $-6 - 8(k - 2) = -10$

Answer: $k = \frac{5}{2}$

Exercise $\PageIndex{16}$

Solve: $4(x - 1)-2=5(2x+3)+6$

Answer

		$.$
Distribute.		$.$
Combine like terms.		$.$
Subtract 4x to get the variables only on the right side since $10>4$.		$.$
Simplify.		$.$
Subtract 21 to get the constants on left.		$.$
Simplify.		$.$
Divide by 6.		$.$
Simplify.		$.$
Check:	$.$
Let $x=-\frac{9}{2}$.	$.$
	$.$
	$.$
	$.$
	$.$

Exercise $\PageIndex{17}$

Solve: $6(p-3)-7=5(4p+3)-12$

Answer: $p = -2$

Exercise $\PageIndex{18}$

Solve: $8(q +1)-5=3(2q-4)-1$

Answer: $q = -8$

Exercise $\PageIndex{19}$

Solve: $10[3 - 8(2s-5)] = 15(40 - 5s)$

Answer

		$.$
Simplify from the innermost parentheses first.		$.$
Combine like terms in the brackets.		$.$
Distribute.		$.$
Add 160s to get the s’s to the right.		$.$
Simplify.		$.$
Subtract 600 to get the constants to the left.		$.$
Simplify.		$.$
Divide.		$.$
Simplify.		$.$
Check:	$.$
Substitute s=−2.	$.$
	$.$
	$.$
	$.$
	$.$
	$.$

Exercise $\PageIndex{20}$

Solve: $6[4−2(7y−1)]=8(13−8y)$.

Answer: $y = -\frac{17}{5}$

Exercise $\PageIndex{21}$

Solve: $12[1−5(4z−1)]=3(24+11z)$.

Answer: $z = 0$

Exercise $\PageIndex{22}$

Solve: $0.36(100n+5)=0.6(30n+15)$.

Answer

		$.$
Distribute.		$.$
Subtract 18n to get the variables to the left.		$.$
Simplify.		$.$
Subtract 1.8 to get the constants to the right.		$.$
Simplify.		$.$
Divide.		$.$
Simplify.		$.$
Check:	$.$
Let n=0.4.	$.$
	$.$
	$.$
	$.$

Exercise $\PageIndex{23}$

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

Answer: $n = 1$

Exercise $\PageIndex{24}$

Solve: $0.15(40m−120)=0.5(60m+12)$.

Answer: $m = -1$

Key Concepts

General Strategy for Solving Linear Equations
1. Simplify each side of the equation as much as possible.
  Use the Distributive Property to remove any parentheses.
  Combine like terms.
2. Collect all the variable terms on one side of the equation.
  Use the Addition or Subtraction Property of Equality.
3. Collect all the constant terms on the other side of the equation.
  Use the Addition or Subtraction Property of Equality.
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.
5. Check the solution.
  Substitute the solution into the original equation.