# 2.4 Use a general strategy to solve linear equations

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By the end of this section, you will be able to:
• Solve equations using a general strategy
• Classify equations

Before you get started, take this readiness quiz.

1. Simplify: $\text{−}\left(a-4\right).$
If you missed this problem, review [link] .
2. Multiply: $\frac{3}{2}\left(12x+20\right)$ .
If you missed this problem, review [link] .
3. Simplify: $5-2\left(n+1\right)$ .
If you missed this problem, review [link] .
4. Multiply: $3\left(7y+9\right)$ .
If you missed this problem, review [link] .
5. Multiply: $\left(2.5\right)\left(6.4\right)$ .
If you missed this problem, review [link] .

## Solve equations using the general strategy

Until now we have dealt with solving one specific form of a linear equation. 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.

## How to solve linear equations using the general strategy

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

## Solution     Solve: $5\left(x+3\right)=35.$

$x=4$

Solve: $6\left(y-4\right)=-18.$

$y=1$

## 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 to make sure the result is a true statement.

Solve: $\text{−}\left(y+9\right)=8.$

## Solution 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 $y=-17$ .   Solve: $\text{−}\left(y+8\right)=-2.$

$y=-6$

Solve: $\text{−}\left(z+4\right)=-12.$

$z=8$

Solve: $5\left(a-3\right)+5=-10$ .

## Solution 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 $1$ by dividing both sides by $5$ . Simplify. Check: Let $a=0$ .    Solve: $2\left(m-4\right)+3=-1$ .

$m=2$

Solve: $7\left(n-3\right)-8=-15$ .

$n=2$

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

## Solution 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 $m=2$ .    Solve: $\frac{1}{3}\left(6u+3\right)=7-u$ .

$u=2$

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

$x=4$

Solve: $8-2\left(3y+5\right)=0$ .

## Solution Simplify—use the Distributive Property. Combine like terms. Add $2$ to both sides to collect constants on the right. Simplify. Divide both sides by $-6$ . Simplify. Check: Let $y=-\frac{1}{3}.$ Solve: $12-3\left(4j+3\right)=-17$ .

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

Solve: $-6-8\left(k-2\right)=-10$ .

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

Solve: $4\left(x-1\right)-2=5\left(2x+3\right)+6$ .

## Solution 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 $x=-\frac{9}{2}$ .     #### Questions & Answers

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