Solving Linear Equations with Variable Terms on Both Sides

Equations, Inequalities, and Problem Solving

Solving Linear Equations with Variable Terms on Both Sides

Linear equations often contain variable terms on both sides, requiring systematic steps to isolate the variable and solve for its value. The process involves applying properties of equality and algebraic manipulation.

• Step 1: Apply the Distributive Property The distributive property removes parentheses and simplifies expressions. For example, in the equation $2(x - 32) = 10x$, distribute the 2: $2 \times x - 2 \times 32 = 10x$ $2x - 64 = 10x$

• Step 2: Use the Addition Property of Equality Move all variable terms to one side and constant terms to the other by adding or subtracting terms from both sides: $2x - 64 = 10x$ $2x - 64 - 2x = 10x - 2x$ $-64 = 8x$

• Step 3: Use the Multiplication Property of Equality Isolate the variable by dividing both sides by the coefficient of the variable: $\frac{-64}{8} = \frac{8x}{8}$ $-8 = x$

• Step 4: Check the Solution Substitute the solution back into the original equation to verify correctness: $2(x - 32) = 10x$ $2(-8 - 32) = 10(-8)$ $2(-40) = -80$ $-80 = -80$ (True statement)

Key Terms:

• Distributive Property: $a(b + c) = ab + ac$

• Addition Property of Equality: If $a = b$, then $a + c = b + c$

• Multiplication Property of Equality: If $a = b$, then $a \times c = b \times c$ (for $c \neq 0$)

Example: Solve $2(x - 32) = 10x$

1. Distribute: $2x - 64 = 10x$

2. Subtract $2x$: $-64 = 8x$

3. Divide by 8: $x = -8$

4. Check: $2(-8 - 32) = 10(-8) \rightarrow -80 = -80$

The solution set is {$-8$}.