152. Solving Equations with Algebraic Fractions

Learning Intentions

• To understand that fractions are used in algebra to indicate division
• Solve equations involving algebraic fractions

Pre-requisite Summary

• A fraction bar represents division, so means
• An equation states that two expressions are equal
• Solving an equation means finding a value that makes the equation true
• Equivalent equations have the same solution
• Multiplying both sides of an equation by the same non-zero number gives an equivalent equation
• To remove a fraction, it is often helpful to multiply both sides by the denominator
• Solutions should be checked by substitution into the original equation

Worked Examples

Worked Example 1

Solve .

Worked Example 2

Solve .

Worked Example 3

Solve .

Worked Example 4

Solve .

Worked Example 5

Solve .

Worked Example 6

Solve .

Worked Example 7

Solve .

Worked Example 8

Solve .

Problems

Problem 1

Solve .

Problem 2

Solve .

Problem 3

Solve .

Problem 4

Solve .

Problem 5

Solve .

Problem 6

Solve .

Problem 7

Solve .

Problem 8

Solve .

Exercises

Understanding and Fluency

Exercise 1.

Write each fraction as a division statement.

a)

b)

c)

Exercise 2.

Solve each equation.

a)

b)

c)

Exercise 3.

Solve each equation.

a)

b)

c)

Exercise 4.

Solve each equation.

a)

b)

c)

Exercise 5.

Solve each equation.

a)

b)

c)

Exercise 6.

Solve each equation.

a)

b)

c)

Exercise 7.

Solve each equation.

a)

b)

c)

Exercise 8.

Solve and Check each equation.

a)

b)

c)

Reasoning

Exercise 9.

Explain why can be solved by multiplying both sides by .

Exercise 10.

A student solves by dividing both sides by and gets . Explain the error.

Exercise 11.

Explain why the fraction bar in means the whole expression is being divided by .

Exercise 12.

Noah says that means . Explain why Noah is incorrect.

Problem-solving

Exercise 13.

A ribbon of length cm is cut into equal pieces. Each piece is cm long. Write and solve an equation to Solve .

Exercise 14.

A plumber divides a pipe of length metres into equal sections. Each section is metres long. Write and solve an equation to find .

Exercise 15.

The total cost of dollars is shared equally among students, and each student pays $8. Write and solve an equation to find .

Exercise 16.

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

Potential Misunderstandings

• Thinking a fraction in algebra means only “part of a whole” rather than division
• Forgetting that the fraction bar means divide by the denominator
• Treating only part of the numerator as being divided when the whole numerator should be divided
• Dividing both sides by the denominator when the denominator should be cleared by multiplication
• Multiplying one side of the equation by the denominator but forgetting to do the same to the other side
• Solving as instead of first removing the denominator
• Making errors with inverse operations after the fraction has been removed
• Forgetting to check the solution in the original equation
• Confusing with
• Thinking that solving equations with algebraic fractions uses different algebra rules from ordinary equations, rather than the same idea of forming equivalent equations

Next: 153. Solving Equations with Pronumerals on Both Sides