Lesson 2: Solving Multi-Step Equations

Property

To solve an equation with two or more operations, we must isolate the variable on one side of the equation. We undo the operations in reverse order. Typically, we undo addition or subtraction first, before undoing multiplication or division.

Examples

• To solve $4x + 5 = 29$, first subtract 5 from both sides to get $4x = 24$. Then, divide both sides by 4 to find $x = 6$.
• To solve $\frac{y}{3} - 2 = 7$, first add 2 to both sides to get $\frac{y}{3} = 9$. Then, multiply both sides by 3 to find $y = 27$.
• To solve $18 = 6 + 2z$, first subtract 6 from both sides to get $12 = 2z$. Then, divide both sides by 2 to find $z = 6$.

Explanation

Think of it as reversing your morning routine. To get back to the start, you undo the last thing you did first. In equations, this means handling addition or subtraction before dealing with multiplication or division to isolate the variable.

Property

If $a$, $b$, and $c$ are any numbers, then

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

• To simplify $5(x+3)$, you distribute the 5 to both $x$ and 3. This gives $5 \cdot x + 5 \cdot 3$, which simplifies to $5x+15$.
• To simplify $6(2a-4) + 7$, first distribute the 6 to get $12a - 24 + 7$. Then, combine the constant terms to get the final expression $12a - 17$.
• To solve the equation $3(y-2) = 9$, first distribute the 3 to get $3y - 6 = 9$. Add 6 to both sides to get $3y = 15$, then divide by 3 to find $y=5$.

Explanation

Think of this law as 'distributing' the term outside the parentheses to every term inside. It helps break down multiplication with a group of terms, turning one complex multiplication into several simpler ones.