Lesson 2.4: Use a General Strategy to Solve Linear Equations

New Concept

This lesson introduces a powerful, multi-step strategy for solving any linear equation. You'll learn to simplify complex expressions, isolate variables, and classify equations as conditional, identities, or contradictions based on their solutions.

What’s next

Next, we'll break down this strategy step-by-step and apply it to various equations through a series of interactive examples and practice problems.

Property

Step 1. Simplify each side of the equation as much as possible.
Use the Distributive Property to remove any parentheses.
Combine like terms.
Step 2. Collect all the variable terms on one side of the equation.
Use the Addition or Subtraction Property of Equality.
Step 3. Collect all the constant terms on the other side of the equation.
Use the Addition or Subtraction Property of Equality.
Step 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.
Step 5. Check the solution.
Substitute the solution into the original equation to make sure the result is a true statement.

Examples

• To solve $-(a+5) = 15$, first distribute the negative sign to get $-a - 5 = 15$. Add 5 to both sides to get $-a = 20$. Finally, divide by $-1$ to find $a = -20$.

• To solve $\frac{1}{2}(4x-6) = 5-x$, distribute $\frac{1}{2}$ to get $2x - 3 = 5 - x$. Add $x$ to both sides to get $3x - 3 = 5$. Add 3 to both sides to get $3x=8$. The solution is $x = \frac{8}{3}$.

Property

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.

Examples

• The equation $3x + 5 = 14$ is a conditional equation. It is only true when $x = 3$, because $3(3) + 5 = 14$.

• Solve $5(y - 2) = 15$. Distributing gives $5y - 10 = 15$. Adding 10 to both sides gives $5y = 25$, so $y=5$. The equation is true only for $y=5$.

Property

An equation that is true for any value of the variable is called an identity. The solution of an identity is all real numbers.

Examples

• The equation $3(x+2) = 3x+6$ is an identity. If we distribute on the left, we get $3x+6 = 3x+6$. This simplifies to $6=6$, which is always true. The solution is all real numbers.

• Consider $7 + 4(m-1) = 4m + 3$. Simplifying the left side gives $7+4m-4 = 4m+3$, which becomes $4m+3 = 4m+3$. This is an identity, and the solution is all real numbers.

Property

An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.

Examples

• The equation $3x - 5 = 3x + 2$ is a contradiction. If we subtract $3x$ from both sides, we are left with the false statement $-5 = 2$. Therefore, there is no solution.

• Solve $4(y+2) = 4y - 1$. Distributing gives $4y+8 = 4y-1$. Subtracting $4y$ from both sides results in $8 = -1$. This is false, so the equation has no solution.