1-3 Solving Equations in One Variable

Property

We solve an equation by undoing in reverse order the operations performed on the variable. We can think of any string of terms as a sum, where the $+$ and $-$ symbols tell us the sign of the term that follows. To solve, first add or subtract terms to isolate the variable term, then multiply or divide to find the variable's value.

Examples

• To solve $12 - 4x = -8$, first subtract 12 from both sides to get $-4x = -20$. Then divide by $-4$ to find $x = 5$.
• To solve $-3y + 7 = 22$, first subtract 7 from both sides to get $-3y = 15$. Then divide by $-3$ to get $y = -5$.
• To solve $\frac{x-5}{3} = -2$, first multiply both sides by 3 to get $x-5 = -6$. Then add 5 to both sides to find $x = -1$.

Explanation

Solving an equation is like being a detective to find the variable's hidden value. You reverse the equation's steps, undoing each operation one by one until the variable is alone on one side of the equals sign.

Property

When equations contain symbols of inclusion such as parentheses, use the Distributive Property to eliminate them first. Then combine like terms and apply inverse operations to solve.

Examples

• Solve $a + 4(2a + 3) = 39$. Distribute the 4: $a + 8a + 12 = 39$. Combine like terms: $9a + 12 = 39$. Solve: $9a = 27$, so $a = 3$.
• To solve $6(c - 3) = 24$, first distribute the 6: $6c - 18 = 24$. Then add 18 to both sides: $6c = 42$. The solution is $c = 7$.
• In $3(x + 5) + 2x = 35$, distribute the 3 to get $3x + 15 + 2x = 35$. Combine terms: $5x + 15 = 35$. Solve to find $x = 4$.

Explanation

Parentheses in an equation are like a locked treasure chest! You have to 'distribute' the number outside to every item inside to unlock it. This unwraps the equation, letting you combine terms and solve for the hidden variable. Multiplying everything inside the parentheses is the key that opens up the problem.

Property

When solving a linear equation, one of three outcomes is possible:

One Solution: The variable isolates to a single value, e.g., $x = a$.