Lesson 3-3: Solve Multi-Step Equations

Property

Addition and Subtraction Properties of Equality:
If the same quantity is added to or subtracted from both sides of an equation, the solution is unchanged.
In symbols, If $a = b$, then $a + c = b + c$ and $a - c = b - c$.
Multiplication and Division Properties of Equality:
If both sides of an equation are multiplied or divided by the same nonzero quantity, the solution is unchanged.
In symbols, If $a = b$, then $ac = bc$ and $\frac{a}{c} = \frac{b}{c}$, $c \neq 0$.

Examples

• If you have the equation $x - 5 = 10$, you can add 5 to both sides to get $x - 5 + 5 = 10 + 5$, which simplifies to $x = 15$.
• For the equation $4y = 28$, you can divide both sides by 4 to get $\frac{4y}{4} = \frac{28}{4}$, which simplifies to $y = 7$.
• If $z + 9 = 12$, you can subtract 9 from both sides to get $z + 9 - 9 = 12 - 9$, which simplifies to $z = 3$.

Explanation

To keep an equation balanced, you must perform the exact same operation on both sides. Whatever you add, subtract, multiply, or divide on one side, you must do to the other side too.

Property

Before you can begin moving terms across the inequality symbol, you must simplify each side independently. This is the "cleanup" phase:

• Step 1: Use the Distributive Property to remove any parentheses.
• Step 2: After distributing, combine any like terms on the same side of the inequality to finish simplifying the expression.

Examples

• Example 1 (Distributing): Simplify the inequality $8 - 2(x + 3) \leq 14$.

First, distribute the -2 to get $8 - 2x - 6 \leq 14$.
Then, combine the constant like terms (8 and -6) to get $-2x + 2 \leq 14$. Now it is ready to solve!

• Example 2 (Multiple Distributions): Simplify $4(x - 8) - (x + 3) > 10$.

Distribute the 4 and the negative sign: $4x - 32 - x - 3 > 10$.
Combine like terms ($4x$ with $-x$, and -32 with -3) to get $3x - 35 > 10$.

• Example 3: Simplify $4(3n + 7) + 5(n - 2) < 50$.

Distribute to get $12n + 28 + 5n - 10 < 50$.
Combine like terms to get $17n + 18 < 50$.

Property

Steps for Solving Linear Equations.

1. Use the distributive law to remove any parentheses.
2. Combine like terms on each side of the equation.
3. By adding or subtracting the same quantity on both sides of the equation, get all the variable terms on one side and all the constant terms on the other.
4. Divide both sides by the coefficient of the variable to obtain an equation of the form $x = a$.

Examples

• Solve $3(x-4) = 9$. First, distribute the 3 to get $3x - 12 = 9$. Add 12 to both sides to get $3x = 21$. Finally, divide by 3 to find $x = 7$.
• Solve $5(y+1) = 2y - 4$. Distribute to get $5y+5 = 2y-4$. Subtract $2y$ from both sides, then subtract 5 from both sides to get $3y = -9$. Divide by 3 to find $y=-3$.
• Solve $25 - 4x = 2x - 5(2-x)$. Distribute to get $25 - 4x = 2x - 10 + 5x$. Combine like terms to get $25 - 4x = 7x - 10$. Add $4x$ to both sides, then add 10 to both sides to get $35 = 11x$, so $x = \frac{35}{11}$.

Explanation

When an equation has parentheses, first use the distributive law to clear them. After that, tidy up by combining like terms on each side. This simplifies the equation, making it easier to isolate the variable and find your solution.