079. Solving Equations with Opposite Operations

Learning Intentions

• To understand that we can find simpler equations by using opposite operations (e.g. dividing by $3$ when the pronumeral was multiplied by $3$)
• 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) $x+5=12$
ii) $3x=18$
iii) $x-7=4$

Worked Example 2

Solve each one-step equation algebraically using equivalent equations:
a) $x+6=15$
b) $x-9=4$
c) $5x=35$

Worked Example 3

Solve each one-step equation algebraically using equivalent equations:
a) ${\displaystyle \frac{x}{4}}=7$
b) $-3x=18$
c) $x+13=-2$

Worked Example 4

Solve each two-step equation algebraically using equivalent equations:
a) $2x+3=11$
b) $4x-5=19$
c) ${\displaystyle \frac{x}{3}}+2=8$

Worked Example 5

Solve each two-step equation algebraically using equivalent equations:
a) $3x+7=22$
b) $5x-4=16$
c) $2x-9=-1$

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 $3x+5=20$

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) $x+8=14$
ii) $4x=24$
iii) $x-6=10$

Problem 2

Solve each one-step equation algebraically using equivalent equations:
a) $x+7=18$
b) $x-5=9$
c) $6x=42$

Problem 3

Solve each one-step equation algebraically using equivalent equations:
a) ${\displaystyle \frac{x}{5}}=6$
b) $-2x=14$
c) $x+11=-3$

Problem 4

Solve each two-step equation algebraically using equivalent equations:
a) $2x+4=12$
b) $3x-2=13$
c) ${\displaystyle \frac{x}{2}}+3=9$

Problem 5

Solve each two-step equation algebraically using equivalent equations:
a) $4x+6=26$
b) $5x-7=18$
c) $3x-8=7$

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 $2x+9=21$

Exercises

Understanding and Fluency

1. State the opposite operation for each:
   a) add $4$
   b) subtract $9$
   c) multiply by $6$
   d) divide by $3$

2. State the operation needed to undo each step:
   a) $x+5$
   b) $x-7$
   c) $4x$
   d) ${\displaystyle \frac{x}{8}}$

3. Solve each one-step equation:
   a) $x+3=10$
   b) $x-4=9$
   c) $2x=14$

4. Solve each one-step equation:
   a) ${\displaystyle \frac{x}{3}}=5$
   b) $7x=49$
   c) $x+12=20$

5. Solve each one-step equation:
   a) $x-11=6$
   b) $-4x=20$
   c) $x+8=-1$

6. Solve each two-step equation:
   a) $2x+1=9$
   b) $3x+4=19$
   c) $4x-3=13$

7. Solve each two-step equation:
   a) $5x-2=18$
   b) $2x-7=5$
   c) ${\displaystyle \frac{x}{4}}+1=6$

8. Solve each two-step equation:
   a) ${\displaystyle \frac{x}{5}}-2=3$
   b) $6x+5=35$
   c) $3x-10=11$

Reasoning

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

10. A student solves $x+6=14$ by subtracting $6$ from only the left-hand side and writes $x=14$. Explain the mistake.

11. Explain why solving $3x=21$ uses division rather than subtraction.

12. A student solves $2x+5=13$ by dividing both sides by $2$ first. Explain why this does not simplify the equation correctly.

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

14. A student says that if $x=4$ solves $2x+3=11$, there is no need to substitute into both sides. Explain why checking is still useful.

Problem-solving

15. A phone plan cost is modelled by $3x+10=28$, where $x$ is the number of extra gigabytes used. Solve the equation and check the solution.

16. A taxi fare is modelled by $5x+4=24$, where $x$ is the number of kilometres travelled. Solve the equation and check the solution.

17. A student’s score is modelled by ${\displaystyle \frac{x}{2}}+6=15$. Solve the equation and check the solution.

18. A game score changes according to $4x-7=21$. Solve the equation and check the solution.

19. A ribbon length problem is modelled by $2x+8=26$. Solve the equation and check the solution.

20. A container problem is modelled by $3x-5=16$. 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