079. Solving Equations with Opposite Operations

Learning Intentions

• To understand that we can Solve simpler equations by Use opposite operations (e.g. dividing by when the pronumeral was multiplied by )
• Solve one-step equations algebraically (using equivalent equations)
• solve two-step equations algebraically (using equivalent equations)
• To understand that solutions can be checked by substitution into both sides of an equation

Pre-requisite Summary

• Understand that an equation states that two expressions are equal
• Know that equivalent equations have the same solution
• Understand that opposite operations undo each other, such as addition and subtraction, or multiplication and division
• Be able to Apply the same operation to both sides of an equation
• Know that a solution is a value that makes the equation true
• Be able to Substitute a value into an expression and Evaluate it
• Recall order of operations when simplifying each side after substitution
• Be able to work accurately with integers and simple fractions if they appear in equations

Worked Examples

Worked Example 1

a) Explain why subtraction is the opposite operation of addition.

b) Explain why division is the opposite operation of multiplication.

c) State the opposite operation needed to Simplify each equation:

i)

ii)

iii)

Worked Example 2

Solve each one-step equation algebraically using equivalent equations:

a)

b)

c)

Worked Example 3

Solve each one-step equation algebraically using equivalent equations:

a)

b)

c)

Worked Example 4

Solve each two-step equation algebraically using equivalent equations:

a)

b)

c)

Worked Example 5

Solve each two-step equation algebraically using equivalent equations:

a)

b)

c)

Worked Example 6

For each equation:

a) solve the equation algebraically

b) Check the solution by substitution into both sides

c) state whether the solution is correct

For

Problems

Problem 1

a) Explain why subtraction is the opposite operation of addition.

b) Explain why division is the opposite operation of multiplication.

c) State the opposite operation needed to simplify each equation:

i)

ii)

iii)

Problem 2

Solve each one-step equation algebraically using equivalent equations:

a)

b)

c)

Problem 3

Solve each one-step equation algebraically using equivalent equations:

a)

b)

c)

Problem 4

Solve each two-step equation algebraically using equivalent equations:

a)

b)

c)

Problem 5

Solve each two-step equation algebraically using equivalent equations:

a)

b)

c)

Problem 6

For each equation:

a) solve the equation algebraically

b) check the solution by substitution into both sides

c) state whether the solution is correct

For

Exercises

Understanding and Fluency

Exercise 1.

State the opposite operation for each:

a) add

b) subtract

c) multiply by

d) divide by

Exercise 2.

State the operation needed to undo each step:

a)

b)

c)

d)

Exercise 3.

Solve each one-step equation:

a)

b)

c)

Exercise 4.

Solve each one-step equation:

a)

b)

c)

Exercise 5.

Solve each one-step equation:

a)

b)

c)

Exercise 6.

Solve each two-step equation:

a)

b)

c)

Exercise 7.

Solve each two-step equation:

a)

b)

c)

Exercise 8.

Solve each two-step equation:

a)

b)

c)

Reasoning

Exercise 9.

Explain why the same operation must be applied to both sides of an equation.

Exercise 10.

A student solves by subtracting from only the left-hand side and writes . Explain the mistake.

Exercise 11.

Explain why solving uses division rather than subtraction.

Exercise 12.

A student solves by dividing both sides by first. Explain why this does not simplify the equation correctly.

Exercise 13.

Explain why checking by substitution confirms whether a solution is correct.

Exercise 14.

A student says that if solves , there is no need to substitute into both sides. Explain why checking is still useful.

Problem-solving

Exercise 15.

A phone plan cost is modelled by , where is the number of extra gigabytes used. Solve the equation and check the solution.

Exercise 16.

A taxi fare is modelled by , where is the number of kilometres travelled. Solve the equation and check the solution.

Exercise 17.

A student’s score is modelled by . Solve the equation and check the solution.

Exercise 18.

A game score changes according to . Solve the equation and check the solution.

Exercise 19.

A ribbon length problem is modelled by . Solve the equation and check the solution.

Exercise 20.

A container problem is modelled by . Solve the equation and check the solution.

Potential Misunderstandings

• Students may apply an opposite operation to only one side of an equation
• Students may Use the wrong opposite operation, such as subtracting instead of dividing
• Students may forget the order needed in two-step equations and try to undo multiplication before undoing addition or subtraction
• Students may divide incorrectly when the pronumeral has a coefficient
• Students may make sign errors when adding or subtracting negative numbers
• Students may think solving means “moving numbers across” without understanding that equivalent equations are being formed
• Students may substitute into only one side when checking a solution
• Students may stop after finding a value and not Verify that it makes both sides equal

Next: 080e. Solving Equations Involving Fractions