151r. Reviewing Equivalent Equations and Solving Algebraically

151. Reviewing Equivalent Equations and Solving Algebraically

Learning Intentions

• To understand what it means for two equations to be equivalent
• Solve equivalent equations by applying the same operation to both sides
• Solve one-step and two-step equations algebraically by finding equivalent equations

Pre-requisite Summary

• Know that an equation is a mathematical statement with an equals sign
• Understand that a solution makes an equation true
• Be able to solve simple equations by inspection
• Know the four basic operations: addition, subtraction, multiplication and division
• Understand that the same operation can be performed to equal quantities without changing equality
• Be able to Substitute a value back into an equation to check a solution
• Know that inverse operations undo each other

Worked Examples

Worked Example 1

State whether each pair of equations is equivalent:

a) and

b) and

c) and

Worked Example 2

Find an equivalent equation by applying the same operation to both sides:

a)

b)

c)

Worked Example 3

Solve each one-step equation algebraically:

a)

b)

c)

Worked Example 4

Solve each one-step equation algebraically:

a)

b)

c)

Worked Example 5

Solve each two-step equation algebraically:

a)

b)

c)

Worked Example 6

Solve each equation and check by substitution:

a)

b)

c)

Problems

Problem 1

State whether each pair of equations is equivalent:

a) and

b) and

c) and

Problem 2

Find an equivalent equation by applying the same operation to both sides:

a)

b)

c)

Problem 3

Solve each one-step equation algebraically:

a)

b)

c)

Problem 4

Solve each one-step equation algebraically:

a)

b)

c)

Problem 5

Solve each two-step equation algebraically:

a)

b)

c)

Problem 6

Solve each equation and check by substitution:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Complete each statement:

a) Two equations are equivalent if they have the same ______

b) To keep an equation equivalent, Apply the same operation to ______ sides

c) Solving algebraically means finding simpler ______ equations

Exercise 2.

State whether each pair of equations is equivalent:

a) and

b) and

c) and

d) and

Exercise 3.

Find an equivalent equation by applying the same operation to both sides:

a)

b)

c)

d)

Exercise 4.

Solve each one-step equation algebraically:

a)

b)

c)

d)

Exercise 5.

Solve each one-step equation algebraically:

a)

b)

c)

d)

Exercise 6.

Solve each two-step equation algebraically:

a)

b)

c)

d)

Exercise 7.

Solve each two-step equation algebraically:

a)

b)

c)

d)

Exercise 8.

Solve and check by substitution:

a)

b)

c)

d)

Reasoning

Exercise 9.

Explain what it means for two equations to be equivalent.

Exercise 10.

A student says that if you add to one side of an equation, the equation stays equivalent without changing the other side. Explain the mistake.

Exercise 11.

Noah says that and are equivalent because both equations have . Is he correct? Explain.

Exercise 12.

Explain why solving starts by subtracting from both sides.

Exercise 13.

A student solves by dividing only the left side by . Describe the error.

Problem-solving

Exercise 14.

A game score is modelled by the equation . Solve the equation to find .

Exercise 15.

A student buys notebooks and then pays a $ fee, for a total of $ . Write and solve an equation to find the cost of one notebook.

Exercise 16.

A rope is cut into equal pieces, and each piece is then shortened by m to give a final length of m. Write and solve an equation for the original piece length.

Exercise 17.

A phone plan charges a fixed fee of $ plus $ per gigabyte. The total bill is $ . Write and solve an equation to find the number of gigabytes used.

Exercise 18.

A number is divided by and then increased by to give . Write and solve an equation to find the number.

Exercise 19.

A bag contains some marbles. After doubling the number and subtracting , the result is . Write and solve an equation to find the number of marbles.

Potential Misunderstandings

• Students may think equivalent equations only need to look similar
• Students may forget that equivalent equations must have the same solution
• Students may apply an operation to only one side of an equation
• Students may use the wrong inverse operation when solving
• Students may solve one-step equations correctly but not two-step equations in the correct order
• Students may forget to undo addition or subtraction before undoing multiplication or division
• Students may make arithmetic errors when simplifying each side
• Students may not check a solution by substitution

Next: 152. Solving Equations with Algebraic Fractions