078. Equivalent Equations

Learning Intentions

• To understand what it means for two equations to be equivalent
• apply an operation to both sides of an equation to form an equivalent equation
• determine that two equations are equivalent by finding an operation that has been applied to both sides

Pre-requisite Summary

• Understand that an equation states that two expressions are equal
• Know that a solution to an equation is a value that makes the equation true
• Be able to substitute values into an equation to test whether it is true
• Recall the operations addition, subtraction, multiplication and division
• Understand that the same operation can be applied to equal quantities without changing equality
• Be able to simplify simple expressions after carrying out an operation
• Know that equations can look different but still have the same solution set

Worked Examples

Worked Example 1

a) Explain what it means for the equations $x+3=8$ and $x=5$ to be equivalent.
b) State the solution of each equation.
c) Explain why the equations are equivalent.

Worked Example 2

Apply an operation to both sides to form an equivalent equation:
a) $x+4=11$
b) $y-6=9$
c) state the operation used in each case

Worked Example 3

Apply an operation to both sides to form an equivalent equation:
a) $2x=14$
b) ${\displaystyle \frac{m}{3}}=5$
c) state the operation used in each case

Worked Example 4

Determine whether two equations are equivalent by identifying the operation applied to both sides:
a) $x+7=12$ and $x=5$
b) $3p=18$ and $p=6$
c) $q-4=9$ and $q=13$

Worked Example 5

Determine whether the second equation is equivalent to the first:
a) $2x+3=11$ and $2x=8$
b) $5y-1=19$ and $5y=20$
c) explain the operation applied to both sides

Worked Example 6

For each pair of equations:
a) decide whether they are equivalent
b) identify the operation applied to both sides, if any
c) state the common solution
For $4x=20$ and $x=5$

Problems

Problem 1

a) Explain what it means for the equations $x+5=12$ and $x=7$ to be equivalent.
b) State the solution of each equation.
c) Explain why the equations are equivalent.

Problem 2

Apply an operation to both sides to form an equivalent equation:
a) $x+6=14$
b) $y-8=7$
c) state the operation used in each case

Problem 3

Apply an operation to both sides to form an equivalent equation:
a) $3x=21$
b) ${\displaystyle \frac{m}{4}}=6$
c) state the operation used in each case

Problem 4

Determine whether two equations are equivalent by identifying the operation applied to both sides:
a) $x+9=15$ and $x=6$
b) $4p=24$ and $p=6$
c) $q-7=5$ and $q=12$

Problem 5

Determine whether the second equation is equivalent to the first:
a) $2x+4=12$ and $2x=8$
b) $6y-2=22$ and $6y=24$
c) explain the operation applied to both sides

Problem 6

For each pair of equations:
a) decide whether they are equivalent
b) identify the operation applied to both sides, if any
c) state the common solution
For $5x=35$ and $x=7$

Problems

Problem 1

a) Are $x+2=9$ and $x=7$ equivalent?
b) Explain using solutions.

Problem 2

a) From $x+8=15$, form an equivalent equation by applying one operation to both sides.
b) Name the operation.

Problem 3

a) From $4x=28$, form an equivalent equation by applying one operation to both sides.
b) Name the operation.

Problem 4

a) Are $y-3=10$ and $y=13$ equivalent?
b) What operation connects them?

Problem 5

a) Are $2m+5=17$ and $2m=12$ equivalent?
b) What operation connects them?

Problem 6

a) Are ${\displaystyle \frac{p}{5}}=4$ and $p=20$ equivalent?
b) What operation connects them?

Exercises

Understanding and Fluency

1. State whether each pair of equations is equivalent:
   a) $x+4=10$ and $x=6$
   b) $y-5=9$ and $y=14$
   c) $m+3=7$ and $m=5$

2. State whether each pair of equations is equivalent:
   a) $2x=16$ and $x=8$
   b) $3y=15$ and $y=6$
   c) ${\displaystyle \frac{p}{4}}=3$ and $p=12$

3. Form an equivalent equation by applying an operation to both sides:
   a) $x+7=13$
   b) $y-4=11$
   c) $m+9=20$

4. Form an equivalent equation by applying an operation to both sides:
   a) $2x=18$
   b) $5y=25$
   c) ${\displaystyle \frac{q}{3}}=7$

5. Identify the operation that has been applied to both sides:
   a) $x+6=14\to x=8$
   b) $y-2=9\to y=11$
   c) $3m=21\to m=7$

6. Identify the operation that has been applied to both sides:
   a) $2p+1=9\to 2p=8$
   b) $4q-3=13\to 4q=16$
   c) ${\displaystyle \frac{r}{5}}=6\to r=30$

7. Decide whether the equations are equivalent and state the common solution if they are:
   a) $x+1=5$ and $x=4$
   b) $2y=12$ and $y=5$
   c) $z-8=3$ and $z=11$

8. Decide whether the equations are equivalent and state the common solution if they are:
   a) $5m=35$ and $m=7$
   b) ${\displaystyle \frac{n}{2}}=9$ and $n=18$
   c) $p+6=10$ and $p=5$

Reasoning

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

10. A student says two equations are equivalent if they look similar. Explain why this is incorrect.

11. Explain why adding the same number to both sides of an equation gives an equivalent equation.

12. A student changes $x+4=9$ into $x=9-4$ and says this is not an equivalent equation because it looks different. Explain the mistake.

13. Explain why applying different operations to the two sides of an equation does not usually produce an equivalent equation.

14. A student says $2x=14$ and $x=14-2$ are equivalent. Explain why this is incorrect.

Problem-solving

15. A student starts with the equation $x+12=20$.
    a) Form an equivalent equation with $x$ alone on one side.
    b) State the operation used.
    c) State the solution.

16. A balance puzzle is modelled by $3m=27$.
    a) Form an equivalent equation.
    b) State the operation used on both sides.
    c) State the solution.

17. A ticket problem is modelled by $2t+4=18$.
    a) Form an equivalent equation by removing the constant term.
    b) State the operation used.
    c) Decide whether the new equation is equivalent to the original.

18. A container problem is modelled by ${\displaystyle \frac{c}{6}}=5$.
    a) Form an equivalent equation with $c$ alone.
    b) State the operation used.
    c) State the solution.

Potential Misunderstandings

• Students may think equivalent equations must look the same
• Students may think two equations are equivalent only if they have the same numbers in them
• Students may apply an operation to only one side of an equation
• Students may apply different operations to the two sides and still expect the equations to remain equivalent
• Students may confuse forming an equivalent equation with simply calculating one side
• Students may forget that equivalent equations must have the same solution set
• Students may divide or multiply incorrectly when the pronumeral has a coefficient
• Students may think a rearranged equation is different in meaning just because its appearance changes

Next: 079. Solving Equations with Opposite Operations