077. Solving Equations by Inspection

Learning Intentions

• To understand that a solution to an equation is the value of the pronumeral that makes the equation true
• use inspection (also called trial and error) to find a solution

Pre-requisite Summary

• Understand the difference between an expression and an equation
• Know that an equation is true when both sides have the same value
• Be able to substitute a number for a pronumeral
• Be able to evaluate simple expressions after substitution
• Understand that a pronumeral can stand for an unknown number
• Recall basic number facts for addition, subtraction, multiplication and division
• Be able to check whether a statement is true or false
• Understand that “inspection” means trying sensible values and checking them

Worked Examples

Worked Example 1

a) Explain what it means for a number to be a solution to the equation $x+4=9$.
b) Use inspection to find the solution.
c) Check the solution by substitution.

Worked Example 2

Use inspection to find the solution:
a) $x-3=5$
b) $x+7=12$
c) $x-8=1$

Worked Example 3

Use inspection to find the solution:
a) $2x=10$
b) $3x=15$
c) $5x=20$

Worked Example 4

Use inspection to find the solution:
a) ${\displaystyle \frac{x}{2}}=4$
b) ${\displaystyle \frac{x}{3}}=5$
c) ${\displaystyle \frac{x}{5}}=6$

Worked Example 5

Use inspection to find the solution:
a) $2x+1=7$
b) $3x-2=10$
c) $4x+3=15$

Worked Example 6

For each equation:
a) guess a value for the pronumeral
b) test it in the equation
c) adjust if needed and find the solution
For $2x+5=13$

Problems

Problem 1

a) Explain what it means for a number to be a solution to the equation $x+6=11$.
b) Use inspection to find the solution.
c) Check the solution by substitution.

Problem 2

Use inspection to find the solution:
a) $x-4=6$
b) $x+9=14$
c) $x-7=2$

Problem 3

Use inspection to find the solution:
a) $2x=12$
b) $4x=16$
c) $6x=24$

Problem 4

Use inspection to find the solution:
a) ${\displaystyle \frac{x}{2}}=6$
b) ${\displaystyle \frac{x}{4}}=3$
c) ${\displaystyle \frac{x}{5}}=4$

Problem 5

Use inspection to find the solution:
a) $2x+3=9$
b) $3x-1=11$
c) $5x+2=17$

Problem 6

For each equation:
a) guess a value for the pronumeral
b) test it in the equation
c) adjust if needed and find the solution
For $3x+4=16$

Exercises

Understanding and Fluency

1. State whether the given value is a solution:
   a) $x+3=8$, when $x=5$
   b) $x-2=6$, when $x=3$
   c) $2x=14$, when $x=7$

2. State whether the given value is a solution:
   a) $3x=12$, when $x=5$
   b) ${\displaystyle \frac{x}{4}}=2$, when $x=8$
   c) $2x+1=9$, when $x=4$

3. Use inspection to find the solution:
   a) $x+5=12$
   b) $x-6=3$
   c) $x+8=15$

4. Use inspection to find the solution:
   a) $2x=8$
   b) $3x=21$
   c) $4x=20$

5. Use inspection to find the solution:
   a) ${\displaystyle \frac{x}{2}}=3$
   b) ${\displaystyle \frac{x}{3}}=4$
   c) ${\displaystyle \frac{x}{6}}=2$

6. Use inspection to find the solution:
   a) $2x+1=5$
   b) $3x+2=11$
   c) $4x-1=7$

7. Use inspection to find the solution:
   a) $2x+5=11$
   b) $5x-3=12$
   c) $3x+4=19$

8. Find the solution and check it by substitution:
   a) $x-9=4$
   b) $5x=25$
   c) ${\displaystyle \frac{x}{7}}=3$

Reasoning

9. Explain why a solution to an equation must make both sides equal.

10. A student says that $x=4$ is a solution to $2x+1=10$ because $2+4+1=7$. Explain the mistake.

11. Explain why substitution is useful for checking a solution found by inspection.

12. A student tries only one value, gets a false statement, and says the equation has no solution. Explain why this is incorrect.

Problem-solving

13. A number increased by $4$ equals $13$. Write an equation and use inspection to find the number.

14. Three times a number is $27$. Write an equation and use inspection to find the number.

15. A number divided by $5$ equals $6$. Write an equation and use inspection to find the number.

16. Twice a number plus $3$ equals $17$. Write an equation and use inspection to find the number.

17. A ticket cost is described by $2x+5=19$, where $x$ is the number of extra rides. Use inspection to find $x$.

18. A game score follows the rule $4x-2=18$. Use inspection to find the score value $x$.

Potential Misunderstandings

• Students may think any value written beside the pronumeral is automatically a solution
• Students may forget that a solution must make the equation true
• Students may substitute incorrectly when checking a guessed value
• Students may confuse $2x$ with $2+x$
• Students may try unreasonable values and not use number facts to guide inspection
• Students may stop after one incorrect guess and assume there is no solution
• Students may find a correct value but not check it in the original equation
• Students may think trial and error means random guessing rather than sensible testing and adjusting

Next: 078. Equivalent Equations