Solving 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.

Example: How to Solve Linear Equations Using the General Strategy

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

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.

In the second row of the table, the first cell says: “Step 2. Collect all variable terms on one side of the equation.” In the second cell, the instructions say: “Nothing to do—all x’s are on the left side. The third cell is blank.

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.

In the fourth row of the table, the first cell says: “Step 4. Make the coefficient of the variable term equal to 1.” In the second cell, the instructions say: “Divide each side by negative 6. Simplify. The third cell contains the same equation divided by negative 6 on both sides: negative 6x over negative 6 equals 42 over negative 6, with “divided by negative 6” written in red on both sides. Below this is the answer to the equation: x equals negative 7.

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.

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.

Example

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 \(y=-17\).

Example

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 \(a=0\).

Example

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 \(m=2\).

Example

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 \(y=-\frac{1}{3}.\)

Example

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 \(x=-\frac{9}{2}\).

Example

Solve: \(10\left[3-8\left(2s-5\right)\right]=15\left(40-5s\right)\).

Solution


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 \(s=-2\).

Example

Solve: \(0.36\left(100n+5\right)=0.6\left(30n+15\right)\).

Solution


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 \(n=0.4\).

Solving Equations Using the General Strategy