Lesson 2: Solving Multi-Step Equations

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.

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

The mean (or average) of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers in the set.