2.4 Use a general strategy to solve linear equations

Elementary algebra Page 1 / 6

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.

Simplify: $− (a - 4) .$
If you missed this problem, review [link] .
Multiply: $\frac{3}{2} (12 x + 20)$ .
If you missed this problem, review [link] .
Simplify: $5 - 2 (n + 1)$ .
If you missed this problem, review [link] .
Multiply: $3 (7 y + 9)$ .
If you missed this problem, review [link] .
Multiply: $(2.5) (6.4)$ .
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: $−6 (x + 3) = 24 .$

Solution

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.


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 $−1 y$ .
Make the coefficient of the variable term to equal to $1$ by dividing both sides by $−1$ .
Simplify.
Check:
Let $y = −17$ .


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 $a = 0$ .


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 $m = 2$ .


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 $y = - \frac{1}{3} .$


Distribute.
Combine like terms.
Subtract $4 x$ 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 $x = - \frac{9}{2}$ .

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2.4 Use a general strategy to solve linear equations

Solve equations using the general strategy

How to solve linear equations using the general strategy

Solution

General strategy for solving linear equations.

Solution

Solution

Solution

Solution

Solution

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