Determining Equations and Solutions with Examples

Key Concepts

• Determine whether a value is a solution of an equation.

• Use substitution to show no values are a solution.

Introduction:

Equation:

• An equation is a mathematical sentence that uses an equal sign to show that two expressions are equal. An equation is true when both sides are equal.
• The equation 12 – 6.7 = 5.3 is true.
• An equation may contain a variable.

Solution of an equation:

• A solution of an equation is a value of the variable that makes the equation true.
• The solution of the equation 8m = 72 is 9.
• The method to find out if a given value is a solution to an equation, substitute the value for the variable. If both sides of the equation are equal, the value is a solution to the equation.

Bar diagram:

• Each bar represents one of the values. You can use this visual representation to demonstrate the equality.
• For example to demonstrate that 26 and the unknown value together make 100, you can construct a bar graph as follows,

4.1.1 Determine whether a value is a solution of an equation

Example 1:

Carol received a $21.00 gift card for phone apps. She has used $9.00 of the value and wants to buy one more app from the list below to use up the balance. Which app should Carol buy?

Solution:

Draw a bar diagram and write an equation to show how the quantities are related.

Example 2:           

A pan balance has 3 cubes on one pan and 9 cubes on the other pan. Shirley thinks she should add 5, 6, 7, or 8 cubes to make the pans balance. Find the number of cubes Shirley should add?

Solution:

Draw a bar diagram and write an equation to show how the quantities are related.

4.1.2 Use substitution to show no values are a solution

Example 3:

Russell has a total of y marbles and 4 boxes. He puts 13 marbles in each box and has none left over.

1. Which of his friends, if any, correctly guessed how many marbles Russell has in all?

Solution:

Draw a bar diagram and write an equation to show how the quantities are related.

• How many marbles Russell has in all?

Solution:

Draw a bar diagram and write an equation to show how the quantities are related. 

There are 4 boxes and 13 marbles in each.

4 × 13 = 52, So 52 ÷ 4 = 13

Russell has 52 marbles in all.

Example 4:

Natalie’s family is driving 257 miles to visit Tallahassee. After driving for a while, they pass a sign that reads “Tallahassee: 125 miles.”

Natalie’s younger brother said “We have already driven 102 miles”.

Natalie’s elder brother said “We have already driven 112 miles”.

Natalie’s younger sister said “We have already driven 122 miles”.

Natalie’s mother said “We have already driven 142 miles”.

1. Which of her family members, if any, correctly guessed how many miles the family has already driven?

Solution:

Draw a bar diagram and write an equation to show how the quantities are related.

Of Natalie’s four family members, none correctly guessed the number of miles the family has already driven.

No solution is given in the set of values.

• How many miles the family has already driven?

Solution:

Out of 257 miles of journey, 125 miles are remaining.

257 – 125 = 132, so 132 + 125 = 257.

Natalie’s family has already driven 132 miles.

Exercise:

1. Tell whether the equation 30 + r= 5 is true or false for r= 6. What is the solution of this equation?

2. Karen is going to babysit for 5 hours. She needs to make exactly $20.00 to buy a concert ticket. How much does Karen need to make each hour in order to buy the concert ticket?

3. Substitute the different values of the variable to find the equation. 5 + x = 22 = 7, 12,17, 20

4. Substitute the different values of the variable to find the equation. n- 9 = 33 n = 14, 24, 32,42

5. Substitute the different values of the variable to find the equation. 27 + p =9 p =3,4,6,7

6. Substitute the different values of the variable to find the equation. 81 = 9y

y = 2,5, 8, 9

7. Tell if each equation is true or false for k = 4. a. 32 + k= 8 b. k = 31 – 27

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8. Patrick set up 6 tables for a party. 42 people are coming to the party. Patrick is planning to seat 7 people at each table. Use the equation 42 + p= 6 to explain whether Patrick’s plan will work.

9. There are 28 pennies on one pan of a pan balance and 18 pennies on the other. To make the pans balance, Nicholas thinks 6 pennies should be added to the higher pan. Ryan thinks 9 pennies should be added, and Rachel thinks 10 pennies should be added. Use the equation 28=18 + p to determine who is correct.

10. Cynthia is making a quilt that uses a pattern of triangles like the one shown. Write an equation to represent the missing side length if the perimeter is 19 centimeters.

What have we learned:

• Understand solving an equation as a process.

• Identify equations and variables.

• Use substitution to find solutions to equations.

Concept Map