153. Solving Equations with Pronumerals on Both Sides

Learning Intentions

• To understand that the same term can be subtracted from or added to both sides of an equation to produce an equivalent equation
• solve equations involving pronumerals on both sides

Pre-requisite Summary

• An equation is a statement that two expressions are equal
• Equivalent equations have the same solution
• Adding the same term to both sides of an equation preserves equality
• Subtracting the same term from both sides of an equation preserves equality
• Like terms can be combined to simplify expressions
• Solving an equation means isolating the pronumeral
• Solutions can be checked by substitution into the original equation

Worked Examples

Worked Example 1

Solve $x+3=7$.

Worked Example 2

Solve $x+5=2x$.

Worked Example 3

Solve $3x+4=x+10$.

Worked Example 4

Solve $5a-2=2a+7$.

Worked Example 5

Solve $4m+9=6m-3$.

Worked Example 6

Solve $7p-5=3p+11$.

Worked Example 7

Solve $2y+8=y-4$.

Worked Example 8

Solve $6k+1=2k+13$.

Problems

Problem 1

Solve $x+6=11$.

Problem 2

Solve $x+4=3x$.

Problem 3

Solve $4x+3=x+12$.

Problem 4

Solve $6a-1=2a+11$.

Problem 5

Solve $3m+7=5m-1$.

Problem 6

Solve $8p-4=3p+16$.

Problem 7

Solve $2y+5=y-6$.

Problem 8

Solve $9k+2=4k+17$.

Exercises

Understanding and Fluency

1. Write an equivalent equation by subtracting the same term from both sides.
   a) $x+5=2x$
   b) $3a+4=a+12$
   c) $5m-1=2m+8$

2. Write an equivalent equation by adding the same term to both sides.
   a) $x-7=4$
   b) $2p-5=p+3$
   c) $3y-8=y-2$

3. Solve each equation.
   a) $x+2=9$
   b) $x+4=2x$
   c) $2x+5=x+11$

4. Solve each equation.
   a) $3a+7=a+15$
   b) $4m-2=2m+8$
   c) $6p+1=3p+10$

5. Solve each equation.
   a) $5y+9=2y+18$
   b) $7k-4=3k+12$
   c) $8n+6=5n+21$

6. Solve each equation.
   a) $2x+3=x-5$
   b) $4a+1=6a-7$
   c) $9m-2=5m+14$

7. Solve and check each equation.
   a) $3p+8=p+18$
   b) $6q-5=2q+11$
   c) $4r+7=7r-2$

8. Solve and check each equation.
   a) $2t+9=t+1$
   b) $5b-6=2b+9$
   c) $7c+4=3c+20$

Reasoning

9. Explain why subtracting $x$ from both sides of $x+5=2x$ gives an equivalent equation.

10. A student says that in $3x+4=x+10$, the $x$ terms can be cancelled immediately to give $4=10$. Explain the error.

11. Explain why adding or subtracting the same term from both sides does not change the solution of an equation.

12. Noah solves $5a-2=2a+7$ by subtracting $2a$ from the left side only. Explain why this does not produce an equivalent equation.

Problem-solving

13. Two mobile plans cost the same amount when compared for a certain number of months. One plan costs $x+12$ dollars and the other costs $2x+3$ dollars. Write and solve an equation to find $x$.

14. A student has two expressions for the same perimeter: $4n+6$ and $2n+16$. Write and solve an equation to find $n$.

15. A shop has two discount rules that give the same final price: $6p-5$ dollars and $3p+19$ dollars. Write and solve an equation to find $p$.

16. A builder measures the same length in two ways: $8m-7$ cm and $5m+11$ cm. Write and solve an equation to find $m$.

Potential Misunderstandings

• Thinking a term can be added or subtracted from one side only and still keep the equation balanced
• Believing that moving a term across the equals sign is a separate rule rather than adding or subtracting the same term to both sides
• Forgetting that the whole term must be added or subtracted, including its sign
• Combining unlike terms when simplifying each side
• Cancelling terms incorrectly across the equals sign
• Making sign errors when subtracting a pronumeral term from both sides
• Solving part of the equation correctly but not continuing until the pronumeral is isolated
• Forgetting to check a solution by substitution into the original equation
• Thinking that equations with pronumerals on both sides need a different method from ordinary equations, rather than the same idea of equivalent equations