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Solving Equations Using Algebra

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Common Core For Grade 7

Examples, videos, and solutions to help Grade 7 students learn how to solve equations using Algebra.

New York State Common Core Math Grade 7, Module 2, Lesson 22

Download worksheets for Grade 7, Module 2, Lesson 22

Lesson 22 Student Outcomes

• Students use algebra to solve equations (of the form px + q = r and p(x + q) = r where p, q, and r are specific rational numbers); using techniques of making zero (adding the additive inverse) and making one (multiplying by the multiplicative inverse) to solve for the variable.
• Students identify and compare the sequence of operations used to find the solution to an equation algebraically, with the sequence of operations used to solve the equation with tape diagrams. They recognize the steps as being the same.
• Students solve equations for the value of the variable using inverse operations; by making zero (adding the additive inverse) and making one (multiplying by the multiplicative inverse).

Lesson 22 Summary

• We work backwards to solve an algebraic equation. For example, to find the value of the variable in the equation
6x - 8 = 40:
1. Use the Addition Property of Equality to add the opposite of –8 to each side of the equation to arrive at
6x - 8 + 8 = 40 + 8.
2. Use the Additive Inverse Property to show that -8 + 8 = 0 and so 6x + 0 = 48.
3. Use the Additive Identity Property to arrive at 6x = 48.
4. Then use the Multiplication Property of Equality to multiply both sides of the equation by 1/6 to get:
(1/6)6x = (1/6)48
5. Then use the Multiplicative Inverse Property to show that (1/6)6 = 1 and so 1x = 8
6. Use the Multiplicative Identity Property to arrive at x = 8.

NYS Math Module 2 Grade 7 Lesson 22 Classwork

Example 1: Yoshiro’s New Puppy
Yoshiro has a new puppy. She decides to create an enclosure for her puppy in her back yard. The enclosure is in the shape of a hexagon (six-sided polygon) with one pair of opposite sides running the same distance along the length of two parallel flowerbeds. There are two boundaries at one end of the flowerbeds that are 10 ft. and 12 ft., respectively, and at the other end, the two boundaries are 15 ft. and 20 ft., respectively. If the perimeter of the enclosure is 137 ft., what is the length of each side that runs along the flowerbed?

Question 1: What is the general shape of the puppy yard? Draw a sketch of the puppy yard.

Question 2: Write an equation that would model finding the perimeter of the puppy yard.

Question 3: Model and solve this equation with a tape diagram.

Question 4: Use algebra to solve this equation. First, use the additive inverse to find out what the lengths of the two missing sides are together. Then, use the multiplicative inverse to find the length of one side of the two equal sides.

Example 2: Swim Practice
Jenny is on the local swim team for the summer and has swim practice 4 days per week. The schedule is the same each day. The team swims in the morning and then again for 2 hours in the evening. If she swims 12 hours per week, how long does she swim each morning?
Question 1: Write an algebraic equation to model this problem. Draw a tape diagram to model this problem.
Question 2: Solve the equations algebraically and graphically with the help of the tape diagram.
Question 3: Does your solution make sense in this context? Why?

Exercises 1–5
Solve each equation algebraically, using if-then statements to justify each step.
1. 5x + 4 = 19

2. 15x + 14 = 19

3. Claire's mom found a very good price on a large computer monitor. She paid $325 for a monitor that was only more than half the original price. What was the original price?

4. 2(x + 4) = 18

5. Ben's family left for vacation after his Dad came home from work on Friday. The entire trip was 600 mi. Dad was very tired after working a long day and decided to stop and spend the night in a hotel after 4 hours of driving. The next morning, Dad drove the remainder of the trip. If the average speed of the car was 60 miles per hour, what was the remaining time left to drive on the second part of the trip? Remember: Distance = rate multiplied by time.

Example 2

Example 1 - Example 2
Lesson 22 Problem Set

For each problem below, explain the steps in finding the value of the variable. Then find the value of the variable, showing each step. Write If-then statements to justify each step in solving the equation.
3. 1/3 y - 18 = 2

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