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 Solve 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
b) Use inspection to find the solution.
c) Check the solution by substitution.
Worked Example 2
Use inspection to find the solution:
a)
b)
c)
Worked Example 3
Use inspection to find the solution:
a)
b)
c)
Worked Example 4
Use inspection to find the solution:
a)
b)
c)
Worked Example 5
Use inspection to find the solution:
a)
b)
c)
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
Problems
Problem 1
a) Explain what it means for a number to be a solution to the equation
b) Use inspection to find the solution.
c) Check the solution by substitution.
Problem 2
Use inspection to find the solution:
a)
b)
c)
Problem 3
Use inspection to find the solution:
a)
b)
c)
Problem 4
Use inspection to find the solution:
a)
b)
c)
Problem 5
Use inspection to find the solution:
a)
b)
c)
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
Exercises
Understanding and Fluency
Exercise 1.
State whether the given value is a solution:
a)
b)
c)
Exercise 2.
State whether the given value is a solution:
a)
b)
c)
Exercise 3.
Use inspection to find the solution:
a)
b)
c)
Exercise 4.
Use inspection to find the solution:
a)
b)
c)
Exercise 5.
Use inspection to find the solution:
a)
b)
c)
Exercise 6.
Use inspection to find the solution:
a)
b)
c)
Exercise 7.
Use inspection to find the solution:
a)
b)
c)
Exercise 8.
Find the solution and check it by substitution:
a)
b)
c)
Reasoning
Exercise 9.
Explain why a solution to an equation must make both sides equal.
Exercise 10.
A student says that
Exercise 11.
Explain why substitution is useful for checking a solution found by inspection.
Exercise 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
Exercise 13.
A number increased by
Exercise 14.
Three times a number is
Exercise 15.
A number divided by
Exercise 16.
Twice a number plus
Exercise 17.
A ticket cost is described by
Exercise 18.
A game score follows the rule
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
with - 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