076. Equations and Expressions

Learning Intentions

• To understand the difference between an equation and an expression
• determine if an equation is true or false, substituting for pronumerals if required
• write an equation given an English description

Pre-requisite Summary

• Understand that a pronumeral or variable can represent a number
• Be able to substitute a number for a pronumeral in an algebraic statement
• Recall the meaning of the operation symbols $+$, $-$, $\times$, and $÷$
• Understand that an expression does not include an equals sign
• Understand that an equation states that two quantities are equal
• Be able to evaluate simple expressions using order of operations
• Be able to interpret words such as sum, difference, product, quotient, and equals

Worked Examples

Worked Example 1

a) State whether $3x+5$ is an expression or an equation.
b) State whether $3x+5=17$ is an expression or an equation.
c) Explain the difference between an expression and an equation.

Worked Example 2

Determine whether each equation is true or false when $x=4$:
a) $x+3=7$
b) $2x=7$
c) $3x-1=11$

Worked Example 3

Determine whether each equation is true or false when $a=2$ and $b=5$:
a) $a+b=7$
b) $2a+b=10$
c) $b-a=3$

Worked Example 4

Write an equation for each description:
a) a number $x$ plus $4$ equals $11$
b) three times $y$ equals $15$
c) $9$ less than $p$ is $12$

Worked Example 5

Write an equation for each description and test whether it is true for the given value:
a) twice $m$ plus $1$ equals $9$, when $m=4$
b) $n-6=10$, when $n=15$
c) $5q=20$, when $q=3$

Worked Example 6

For each statement:
a) write it as an equation
b) substitute the given value
c) decide whether the equation is true or false
“$4$ more than $x$ is $13$”, when $x=9$

Problems

Problem 1

a) State whether $5y-2$ is an expression or an equation.
b) State whether $5y-2=18$ is an expression or an equation.
c) Explain the difference between an expression and an equation.

Problem 2

Determine whether each equation is true or false when $x=3$:
a) $x+5=8$
b) $2x=5$
c) $4x-2=10$

Problem 3

Determine whether each equation is true or false when $a=4$ and $b=3$:
a) $a+b=7$
b) $2a+b=12$
c) $a-b=2$

Problem 4

Write an equation for each description:
a) a number $x$ plus $6$ equals $14$
b) four times $y$ equals $24$
c) $7$ less than $p$ is $9$

Problem 5

Write an equation for each description and test whether it is true for the given value:
a) twice $m$ plus $3$ equals $11$, when $m=4$
b) $n-5=12$, when $n=16$
c) $6q=24$, when $q=5$

Problem 6

For each statement:
a) write it as an equation
b) substitute the given value
c) decide whether the equation is true or false
“$5$ more than $x$ is $17$”, when $x=12$

Exercises

Understanding and Fluency

1. State whether each is an expression or an equation:
   a) $2x+7$
   b) $2x+7=15$
   c) $4a-3$

2. State whether each is an expression or an equation:
   a) $y+8=20$
   b) $3m$
   c) $n-4=9$

3. Determine whether each equation is true or false for the given value:
   a) $x+2=9$, when $x=7$
   b) $3x=15$, when $x=4$
   c) $2x+1=11$, when $x=5$

4. Determine whether each equation is true or false for the given value:
   a) $a+b=9$, when $a=4,b=5$
   b) $2a-b=1$, when $a=3,b=4$
   c) $3m+2=14$, when $m=4$

5. Write an equation for each description:
   a) a number $x$ plus $3$ equals $10$
   b) two times $y$ equals $8$
   c) $5$ less than $p$ is $11$

6. Write an equation for each description:
   a) $7$ more than $n$ is $20$
   b) three times $q$ equals $18$
   c) $12$ minus $r$ equals $4$

7. Write an equation, then test whether it is true for the given value:
   a) $4$ more than $x$ is $15$, when $x=11$
   b) twice $y$ is $14$, when $y=7$
   c) $9$ less than $p$ is $3$, when $p=12$

8. Write an equation, then test whether it is true for the given value:
   a) $m+8=17$, when $m=9$
   b) $5n=25$, when $n=6$
   c) $t-7=5$, when $t=12$

Reasoning

9. Explain why $3x+4$ is an expression but $3x+4=16$ is an equation.

10. A student says that $x+5$ is an equation because it has a pronumeral. Explain the mistake.

11. Explain why substituting a value into both sides of an equation helps decide whether it is true or false.

12. A student substitutes $x=4$ into $2x+3=11$ and writes $24+3=11$. Explain the error.

Problem-solving

13. A taxi fare is described by “$6 plus $2 for each kilometre $k$ equals the total fare $14”. Write this as an equation.

14. A student says, “Three times my number is $21$.” Write an equation for this statement.

15. A phone plan cost is described by “$10 plus $3 for each gigabyte $g$ used equals $19”. Write an equation.

16. A rectangle has width $w$. “The width plus $5$ equals $12$.” Write this as an equation and test whether it is true when $w=7$.

17. A game score is described by “Twice the score $s$ minus $1$ equals $9$.” Write the equation and test it for $s=5$.

18. A savings problem says, “$8 more than the amount $x$ saved equals $23.” Write the equation and test it when $x=15$.

Potential Misunderstandings

• Students may think any algebraic statement is an equation, even when there is no equals sign
• Students may think an expression must include a pronumeral, even though numerical expressions also exist
• Students may confuse the equals sign with a signal to calculate rather than a statement that two quantities are equal
• Students may substitute into only one side of an equation when deciding whether it is true or false
• Students may treat $2x$ as $20+x$ or as the two-digit number formed by $2$ and $x$
• Students may reverse word descriptions such as “$5$ less than $p$” and write $5-p$ instead of $p-5$
• Students may think an equation is false just because it is not yet solved

Next: 077. Solving Equations by Inspection