Lesson 3-2: Write and Solve Equations with Variables on Each Side

Property

Steps for Modeling a Problem:

1. Identify the unknown quantity and choose a variable to represent it.
2. Find some quantity that can be expressed in two different ways, and write an equation.
3. Solve the equation, and answer the question in the problem.

Examples

• A recipe's ratio of flour to sugar is 5 to 2. If you use 10 cups of flour, how much sugar do you need? Let $s$ be the number of cups of sugar. The ratio can be written as $\frac{10}{s}$ and as $\frac{5}{2}$. The equation is $\frac{10}{s} = \frac{5}{2}$, so $s=4$ cups of sugar.

• A student and their backpack weigh 145 pounds together. If the student weighs 120 pounds, how much does the backpack weigh? Let $b$ be the backpack's weight. The total weight is both $120+b$ and 145. So, $120+b = 145$, which means the backpack weighs 25 pounds.

Property

Step 1. Choose one side to be the variable side and then the other will be the constant side.
Step 2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
Step 3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
Step 4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
Step 5. Check the solution by substituting it into the original equation.

It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.

Examples

• Given $9x + 3 = 4x + 23$, subtract $4x$ from both sides to get $5x + 3 = 23$. Then subtract 3 to get $5x = 20$, so $x = 4$.