151. Reviewing Equivalent Equations and Solving Algebraically

Learning Intentions

• To understand what it means for two equations to be equivalent
• find 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) $x+3=8$ and $x=5$
b) $2y=10$ and $y=5$
c) $m-4=7$ and $m=3$

Worked Example 2

Find an equivalent equation by applying the same operation to both sides:
a) $x+6=14$
b) $p-5=9$
c) $3a=18$

Worked Example 3

Solve each one-step equation algebraically:
a) $x+7=12$
b) $y-4=9$
c) $5m=20$

Worked Example 4

Solve each one-step equation algebraically:
a) ${\displaystyle \frac{p}{3}}=6$
b) $2q=14$
c) $n+11=15$

Worked Example 5

Solve each two-step equation algebraically:
a) $2x+3=11$
b) $3y-4=14$
c) ${\displaystyle \frac{m}{2}}+5=9$

Worked Example 6

Solve each equation and check by substitution:
a) $4a+1=13$
b) ${\displaystyle \frac{b}{5}}-2=4$
c) $2c-7=9$

Problems

Problem 1

State whether each pair of equations is equivalent:
a) $x+4=9$ and $x=5$
b) $3y=15$ and $y=5$
c) $m-2=8$ and $m=6$

Problem 2

Find an equivalent equation by applying the same operation to both sides:
a) $x+8=17$
b) $p-6=10$
c) $4a=24$

Problem 3

Solve each one-step equation algebraically:
a) $x+5=13$
b) $y-7=6$
c) $6m=24$

Problem 4

Solve each one-step equation algebraically:
a) ${\displaystyle \frac{p}{4}}=5$
b) $3q=18$
c) $n+9=14$

Problem 5

Solve each two-step equation algebraically:
a) $2x+4=16$
b) $4y-3=13$
c) ${\displaystyle \frac{m}{3}}+2=7$

Problem 6

Solve each equation and check by substitution:
a) $5a+2=17$
b) ${\displaystyle \frac{b}{4}}-1=3$
c) $3c-5=10$

Exercises

Understanding and Fluency

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

2. State whether each pair of equations is equivalent:
   a) $x+2=7$ and $x=5$
   b) $2m=12$ and $m=6$
   c) $y-3=10$ and $y=13$
   d) $4p=16$ and $p=5$

3. Find an equivalent equation by applying the same operation to both sides:
   a) $x+9=15$
   b) $a-4=12$
   c) $5m=25$
   d) ${\displaystyle \frac{q}{2}}=7$

4. Solve each one-step equation algebraically:
   a) $x+6=10$
   b) $y-8=5$
   c) $4m=20$
   d) ${\displaystyle \frac{p}{6}}=3$

5. Solve each one-step equation algebraically:
   a) $a+12=19$
   b) $b-9=4$
   c) $7c=35$
   d) ${\displaystyle \frac{d}{5}}=8$

6. Solve each two-step equation algebraically:
   a) $2x+1=9$
   b) $3y+2=14$
   c) $4m-5=11$
   d) ${\displaystyle \frac{p}{2}}+3=8$

7. Solve each two-step equation algebraically:
   a) $5a-4=16$
   b) $2b+7=15$
   c) ${\displaystyle \frac{c}{3}}-2=4$
   d) $6d+1=25$

8. Solve and check by substitution:
   a) $3x+2=11$
   b) $4y-1=15$
   c) ${\displaystyle \frac{m}{4}}+5=8$
   d) $2p-6=10$

Reasoning

9. Explain what it means for two equations to be equivalent.

10. A student says that if you add $3$ to one side of an equation, the equation stays equivalent without changing the other side. Explain the mistake.

11. Noah says that $x+5=12$ and $x=8$ are equivalent because both equations have $x$. Is he correct? Explain.

12. Explain why solving $2x+3=11$ starts by subtracting $3$ from both sides.

13. A student solves $4x=20$ by dividing only the left side by $4$. Describe the error.

Problem-solving

14. A game score is modelled by the equation $2x+5=17$. Solve the equation to find $x$.

15. A student buys $3$ notebooks and then pays a $\mathrm{\$}4$ fee, for a total of $\mathrm{\$}19$. Write and solve an equation to find the cost of one notebook.

16. A rope is cut into $5$ equal pieces, and each piece is then shortened by $2$ m to give a final length of $4$ m. Write and solve an equation for the original piece length.

17. A phone plan charges a fixed fee of $\mathrm{\$}6$ plus $\mathrm{\$}2$ per gigabyte. The total bill is $\mathrm{\$}18$. Write and solve an equation to find the number of gigabytes used.

18. A number is divided by $3$ and then increased by $7$ to give $12$. Write and solve an equation to find the number.

19. A bag contains some marbles. After doubling the number and subtracting $3$, the result is $13$. 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