6-1 Use Substitution to Solve One-Step Equations

Property

An expression is a mathematical phrase that contains numbers, variables, and operations but has no equal sign: $3x + 5$ or $2y - 7$.

An equation is a mathematical sentence that shows two expressions are equal using an equal sign: $3x + 5 = 14$ or $2y - 7 = 11$.

Property

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
To determine whether a number is a solution to an equation:

Step 1. Substitute the number for the variable in the equation.
Step 2. Simplify the expressions on both sides of the equation.
Step 3. Determine whether the resulting equation is true.

• If it is true, the number is a solution.
• If it is not true, the number is not a solution.

Examples

• Is $x=5$ a solution to the equation $3x - 5 = 10$? We substitute $3(5) - 5 = 15 - 5 = 10$. Since $10 = 10$, it is a solution.
• Is $y=-2$ a solution to the equation $4y + 9 = 2y$? We substitute $4(-2) + 9 = -8 + 9 = 1$ on the left, and $2(-2) = -4$ on the right. Since $1 \neq -4$, it is not a solution.
• Is $a = \frac{1}{3}$ a solution to the equation $9a + 2 = 5$? We substitute $9(\frac{1}{3}) + 2 = 3 + 2 = 5$. Since $5=5$, it is a solution.

Property

A problem-solving strategy where you make a reasonable guess, check if it fits the problem's conditions, and revise your guess if it is wrong. Repeat the process until you find the correct answer.

Examples

• Problem: Two numbers add to 20 and their difference is 6. Guess: 10 and 10. Check: $10-10=0$. Revise: 13 and 7. Check: $13+7=20$ and $13-7=6$. Correct!
• Problem: A toy costs 45 cents. You pay with 10 coins (nickels and dimes). Guess: 5 nickels, 5 dimes = 75 cents. Revise: 7 nickels, 3 dimes = 65 cents. Revise again: 9 nickels, 1 dime = 55 cents.

Explanation

Think of yourself as a super sleuth! Make a smart guess to crack the case, check your clues, and if you're wrong, just try another angle. Every guess gets you closer to the solution, detective!