Equations Involving More than One Operation

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Steps to Solving Equations

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Combine Like Terms in Equations

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Combine Like Terms in Expressions

# Equations Involving More than One Operation

Solve the following problem.

Adrian bought three shirts and one pair of pants for a total of \$203. If Adrian paid \$59 for the pants, what was the cost of each shirt?

Let x represent the cost of one shirt. Adrian would have paid 3x + 59 for three shirts and one pair of pants. We need to solve the equation 3x + 59 = 203 for x. This is an equation with two operations: multiplication and division. Which should we eliminate first?

Now that we know the order of operations and the inverse operations, we can solve equations like the above equation that have more than one operation. The key idea to remember is that when solving an equation, we work backwards from the order of operations used to evaluate an expression. The process is like the difference between wrapping or unwrapping a present.

Doing the arithmetic (evaluating an expression) is like wrapping the present.

• We take the present and put it in a box.
• We put the lid on the box.
• We wrap the box in pretty paper.
• We put ribbon around the box,
• and then attach a bow.

Solving an equation is like unwrapping the present to find out what is inside. We do exactly the reverse of what was done to wrap the present.

• We first take off the bow,
• then cut the ribbon,
• take off the wrapping paper,
• remove the lid, and
• finally reach inside and take the present out of the box.

The order of operations tells us to do multiplication before addition or subtraction when doing arithmetic (unless otherwise indicated by parentheses). So when we solve an equation involving both multiplication and addition (or subtraction), we solve in reverse order undoing the addition or subtraction before we undo the multiplication.

Applied to the above example: Solve for the cost of each shirt Adrian bought.

3x + 59 = 203

We first subtract 59 from both sides

(3x + 59) – 59 = 203 – 59.

To obtain

3x = 144.

Next, we divide both sides by 3

(3x) ÷ 3 = 144 ÷ 3.

To obtain

x = 48.

Each shirt would have cost Adrian \$48.

We check the solution by substituting into the left-hand side of the equation and using the order of operations to evaluate the expression to check if we get the right-hand side.

3(48) + 59 = 144 + 59 = 203

## Self-Check Problem

Jan's electric bill for the month of October was \$55. The cost of electricity is 9 cents per kilowattt-hour with a service charge of \$19. How much electricity did Jan use in October?