Solve Equations with Two Operations

Honors Math – Grade 8

Get Ready for the LessonAn alligator hatchling 8 inches long grows about

12 inches per year. The expression 8 + 12a represents the length in inches of an alligator that is a years old. Can you determine the age of the alligator shown?

The alligator is 10 feet 4 inches long, which is the same as 124 inches—(10)(12) + 4.

We can write an equation to solve this problem.

124128 aTo solve equations with more than one operation, often called multi-step equations, undo operations by working backwards.

Solve each equation. Check the solution.

124128 aIsolate the variable on one side of the equation. You need to move the “8” to the other side.

Subtract 8 from both sides.

Divide both sides of the equation by 12.

1241241241168

1243

29128

Remember to check your solution!

Solve each equation.

738 rWhen solving multi-step equations, undo operations by working the order of operations in reverse.

Undo Addition or Subtraction first; then Multiplication or Division.

Subtract 7 from both sides of the equation.

Divide both sides of the equation by 3.

Check your solution mentally.

Solve each equation.

5.45.03

kWhen solving multi-

step equations, undo operations by working the order of operations in reverse.

Undo Addition or Subtraction first; then Multiplication or Division.

Add 0.5 to both sides of the equation.

Multiply both sides of the equation by 3.

Check your solution mentally.

Solve each equation.

69

15

pRemember a fraction bar is a grouping symbol. When solving equations with grouping symbols, use inverse operations to eliminate the denominator first.

Multiply both sides of the equation by 9.

9 9

5415 p1515 39p

The denominator is eliminated and the numerator remains.

Add 15 to both sides of the equation.

Don’t forget to check your solution!

Hugo is buying a pair of water skis that are on sale for 2/3 of the original price. After he uses a $25 gift certificate, the total cost before taxes is $115. What was the original price of the skis?

Define the variable. Let p = the original price

Write and solve an equation for this situation.

2/3 of the price - $25 gift

certificate = $115

When solving multi-step equations, undo operations by working the order of operations in reverse.

Undo Addition or Subtraction first; then Multiplication or Division.

Add 25 to both sides of the equation.

Multiply by the reciprocal on both sides of the equation.

The original price of the skis was $210.

Solve each equation. 3211 bWhen the variable is being subtracted, rewrite the equation using a coefficient of 1.

Undo Addition or Subtraction first; then Multiplication or Division.

Subtract 11 from both sides of the equation.

Divide both sides of the equation by -1.

Check your solution mentally.

Solve each equation.

9

7

9

1

18

w

When an equation contains several fractions, you can either compute with fractions throughout or you can transform the equation into an equivalent equation with integer coefficients.

METHOD 1 – Fractions

Add 1/9 to both sides of the equation.

Multiply both sides of the equation by 18.

METHOD 2 – Use LCD

Multiply EVERY TERM by the LCD.

What is the LCD?

This writes an equivalent equation. Now solve the simpler equation.

Solve each equation.

12

5

12

11

24

z

When an equation contains several fractions, you can either compute with fractions throughout or you can transform the equation into an equivalent equation with integer coefficients.

METHOD 1 – Fractions

Add 11/12 to both sides of the equation.

Multiply both sides of the equation by 24.

METHOD 2 – Use LCD

Multiply EVERY TERM by the LCD.

What is the LCD?

This writes an equivalent equation. Now solve the simpler equation.

Solve each equation.

3598.172 ppIf the unknown appears in more than one term, you will need to combine like terms first.

Remember: Like terms have the same variable raised to the same power!

Remember to group the sign in front of the terms.

Undo Addition or Subtraction first; then Multiplication or Division.

Add 1.98 to both sides

Divide both sides by -5

Be sure to check your solution by substituting the solution into the original equation!

Solve each equation.

1.783.48.17 yyIf the unknown appears in more than one term, you will need to combine like terms first.

Remember: Like terms have the same variable raised to the same power!

Remember to group the sign in front of the terms.

Undo Addition or Subtraction first; then Multiplication or Division.

Subtract 4.3 from both sides

Divide both sides by 5.2

Be sure to check your solution by substituting the solution into the original equation!

Consecutive Integers are integers in counting order, such as 7, 8, and 9.

Beginning with an even integer and counting by two will result in consecutive even integers.

Beginning with an odd integer and counting by two will result in consecutive odd integers.

-4, -2, 0

, 2, 4

, 6, 8

-7, -5, -3, -1, 1, 3

The study of number and the relationships between them is called number theory.

There are three consecutive integers in counting order whose sum is 84. What are the integers?

Define the variables.Let x = the 1st integerx + 1 = the 2nd integerx + 2 = the 3rd integer

Write an equation that shows the sum of the three integers.

Simplify the equation by grouping like terms.

? xxx?21

The three integers are 27, 28, and 29.

Find three consecutive even integers whose sum is -42.

Define the variables.Let n = the 1st even integern + 2 = the 2nd even integern + 4 = the 3rd even integer

Write an equation that shows the sum of the three integers.

4242 nnn

Simplify the equation by grouping like terms.

? nnn?42

The three integers are -16, -14, and -12.

The measure of angle A in triangle ABC is 3 times the measure of angle B, and the measure of angle C, is half the measure of angle B. Find the measure of each angle.

An isosceles triangle has a perimeter of 35 feet. The third side is one-third the length of one of the congruent sides. What are the lengths of the sides of this isosceles triangle?