2. Solving Linear Equations and Inequalities
2.2 Use a General Strategy to Solve Linear Equations
Learning Objectives
By the end of this section it is expected that you will be able to:
- Solve equations using a general strategy
- Classify equations
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.
EXAMPLE 1
Solve:
TRY IT 1
Solve:
Show answer
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 2
Solve:
| Simplify each side of the equation as much as possible by distributing. | |
| The only term is on the left side, so all variable terms are on the left side of the equation. | |
| Add to both sides to get all constant terms on the right side of the equation. | |
| Simplify. | |
| Rewrite as . | |
| Make the coefficient of the variable term to equal to by dividing both sides by . | |
| Simplify. | |
| Check: | |
| Let . | |
TRY IT 2
Solve:
Show answer
EXAMPLE 3
Solve: .
| Simplify each side of the equation as much as possible. | |
| Distribute. | |
| Combine like terms. | |
| The only term is on the left side, so all variable terms are on one side of the equation. | |
| Add 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 by dividing both sides by . | |
| Simplify. | |
| Check: | |
| Let . | |
TRY IT 3
Solve: .
Show answer
EXAMPLE 4
Solve: .
| Distribute. | |
| Add to get the variables only to the left. | |
| Simplify. | |
| Add to get constants only on the right. | |
| Simplify. | |
| Divide by . | |
| Simplify. | |
| Check: | |
| Let . | |
TRY IT 4
Solve: .
Show answer
EXAMPLE 5
Solve: .
| Simplify—use the Distributive Property. | |
| Combine like terms. | |
| Add to both sides to collect constants on the right. | |
| Simplify. | |
| Divide both sides by . | |
| Simplify. | |
| Check: Let |
TRY IT 5
Solve: .
Show answer
EXAMPLE 6
Solve: .
| Distribute. | |
| Combine like terms. | |
| Subtract to get the variables only on the right side since > . | |
| Simplify. | |
| Subtract to get the constants on left. | |
| Simplify. | |
| Divide by 6. | |
| Simplify. | |
| Check: | |
| Let . | |
TRY IT 6
Solve: .
Show answer
EXAMPLE 7
Solve: .
| Simplify from the innermost parentheses first. | |
| Combine like terms in the brackets. | |
| Distribute. | |
| Add to get the s’s to the right. | |
| Simplify. | |
| Subtract 600 to get the constants to the left. | |
| Simplify. | |
| Divide. | |
| Simplify. | |
| Check: | |
| Substitute . | |
TRY IT 7
Solve: .
Show answer
EXAMPLE 8
Solve: .
| Distribute. | |
| Subtract to get the variables to the left. | |
| Simplify. | |
| Subtract to get the constants to the right. | |
| Simplify. | |
| Divide. | |
| Simplify. | |
| Check: | |
| Let . | |
TRY IT 8
Solve: .
Show answer
Classify Equations
When you solve the equation , the solution is . This means the equation is true when we replace the variable, x, with the value . We can show this by checking the solution and evaluating for .
If we evaluate for a different value of x, the left side will not be .
The equation is true when we replace the variable, x, with the value , but not true when we replace x with any other value. Whether or not the equation is true depends on the value of the variable. Equations like this are called conditional equations.
All the equations we have solved so far are conditional equations.
Conditional equation
An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.
Now let’s consider the equation . Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y.
| Distribute. | |
| Subtract to get the ’s to one side. | |
| Simplify—the ’s are gone! |
But is true.
This means that the equation is true for any value of y. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable like this is called an identity.
Identity
An equation that is true for any value of the variable is called an identity.
The solution of an identity is every real number.
What happens when we solve the equation ?
| Subtract to get the constant alone on the right. | |
| Simplify—the ’s are gone! |
But .
Solving the equation led to the false statement . The equation will not be true for any value of z. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a contradiction.
Contradiction
An equation that is false for all values of the variable is called a contradiction.
A contradiction has no solution.
EXAMPLE 9
Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution.
| Distribute. | |
| Combine like terms. | |
| Subtract to get the ’s to one side. | |
| Simplify. | |
| This is a true statement. | The equation is an identity. The solution is every real number. |
TRY IT 9
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:
Show answer
identity; all real numbers
EXAMPLE 10
Classify as a conditional equation, an identity, or a contradiction. Then state the solution.
| Distribute. | |
| Combine like terms. | |
| Add to both sides. | |
| Simplify. | |
| Divide. | |
| Simplify. | |
| The equation is true when . | This is a conditional equation. The solution is |
TRY IT 10
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:
Show answer
conditional equation;
EXAMPLE 11
Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution.
| Distribute. | |
| Combine like terms. | |
| Subtract from both sides. | |
| Simplify. | |
| But . | The equation is a contradiction. It has no solution. |
TRY IT 11
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:
Show answer
contradiction; no solution
Type of equation – Solution
| Type of equation | What happens when you solve it? | Solution |
|---|---|---|
| Conditional Equation | True for one or more values of the variables and false for all other values | One or more values |
| Identity | True for any value of the variable | All real numbers |
| Contradiction | False for all values of the variable | No solution |
Key Concepts
- 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.
- Simplify each side of the equation as much as possible.
Glossary
- conditional equation
- An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.
- contradiction
- An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.
- identity
- An equation that is true for any value of the variable is called an identity. The solution of an identity is all real numbers.
2.2 Exercise Set
In the following exercises, solve each linear equation.
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.
- Coins. Marta 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 .
Answers
- identity; all real numbers
- identity; all real numbers
- conditional equation;
- conditional equation;
- contradiction; no solution
- contradiction; no solution
- conditional equation;
- contradiction; no solution
- identity; all real numbers
- identity; all real numbers
- 8 nickels
Attributions
This chapter has been adapted from “Use a General Strategy to Solve Linear Equations” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Adaptation Statement for more information.