054e. Substituting Integers into Expressions

Learning Intentions

• To understand that substitution involves replacing a pronumeral (letter) with a number
• substitute positive or negative integers into an expression and evaluate

Pre-requisite Summary

• Understand that a pronumeral or variable is a letter that represents a number
• Know that substitution means replacing the letter everywhere it appears with the given value
• Recognise that multiplication can be written without a sign, for example $3x=3\times x$
• Be able to use brackets when substituting negative numbers, for example if $x=-2$, then $3x=3(-2)$
• Recall the order of operations for expressions involving more than one operation
• Be able to add, subtract, multiply and divide positive and negative integers

Worked Examples

Worked Example 1

Substitute and evaluate:
a) Find the value of $x+4$ when $x=3$
b) Find the value of $y-7$ when $y=-2$

Worked Example 2

Substitute and evaluate:
a) Find the value of $3a$ when $a=5$
b) Find the value of $2b$ when $b=-4$

Worked Example 3

Substitute and evaluate:
a) Find the value of $2x+3$ when $x=4$
b) Find the value of $2x+3$ when $x=-4$

Worked Example 4

Substitute and evaluate:
a) Find the value of $5-m$ when $m=2$
b) Find the value of $5-m$ when $m=-3$

Worked Example 5

Substitute and evaluate using order of operations:
a) Find the value of $3p-2$ when $p=-5$
b) Find the value of $4+2q$ when $q=-3$

Worked Example 6

Substitute and evaluate with two pronumerals:
a) Find the value of $2x+y$ when $x=3$ and $y=-4$
b) Find the value of $a-3b$ when $a=-2$ and $b=5$

Problems

Problem 1

Substitute and evaluate:
a) Find the value of $x+6$ when $x=2$
b) Find the value of $y-5$ when $y=-1$

Problem 2

Substitute and evaluate:
a) Find the value of $4a$ when $a=3$
b) Find the value of $3b$ when $b=-6$

Problem 3

Substitute and evaluate:
a) Find the value of $2x-1$ when $x=5$
b) Find the value of $2x-1$ when $x=-5$

Problem 4

Substitute and evaluate:
a) Find the value of $7-n$ when $n=4$
b) Find the value of $7-n$ when $n=-2$

Problem 5

Substitute and evaluate using order of operations:
a) Find the value of $2p+5$ when $p=-4$
b) Find the value of $6-3q$ when $q=-2$

Problem 6

Substitute and evaluate with two pronumerals:
a) Find the value of $x+2y$ when $x=4$ and $y=-3$
b) Find the value of $2a-b$ when $a=-5$ and $b=6$

Exercises

Understanding and Fluency

1. Substitute and evaluate:
   a) $x+3$ when $x=4$
   b) $y-2$ when $y=7$
   c) $m+5$ when $m=-3$

2. Substitute and evaluate:
   a) $2a$ when $a=6$
   b) $3b$ when $b=-2$
   c) $5c$ when $c=-4$

3. Substitute and evaluate:
   a) $x-8$ when $x=5$
   b) $p+9$ when $p=-6$
   c) $4-q$ when $q=-3$

4. Substitute and evaluate:
   a) $2x+1$ when $x=3$
   b) $3y-4$ when $y=-2$
   c) $5+a$ when $a=-7$

5. Substitute and evaluate:
   a) $4m-3$ when $m=2$
   b) $6-2n$ when $n=-4$
   c) $3r+5$ when $r=-1$

6. Substitute and evaluate using two pronumerals:
   a) $x+y$ when $x=5,y=-2$
   b) $2a+b$ when $a=-3,b=4$
   c) $m-2n$ when $m=6,n=-1$

7. Substitute and evaluate using two pronumerals:
   a) $3x-y$ when $x=2,y=-5$
   b) $2p+3q$ when $p=-1,q=4$
   c) $a-b$ when $a=-6,b=-2$

8. Substitute and evaluate using order of operations:
   a) $4+2x$ when $x=-3$
   b) $3y-7$ when $y=5$
   c) $2m+1$ when $m=-8$

Reasoning

9. Explain why $3x$ means $3\times x$ when substituting a value for $x$.

10. A student says that if $x=-4$, then $2x=24$. Explain the mistake.

11. Explain why using brackets helps when substituting a negative number into an expression.

12. A student substitutes $x=-3$ into $5-x$ and gets $2$. Explain why this is incorrect.

Problem-solving

13. The temperature is modelled by $t+4$. Find the value when $t=-6$.

14. A game score is given by $3s-2$. Find the score when $s=-5$.

15. The lift position is represented by $f+7$. Find the value when $f=-3$.

16. A bank balance is modelled by $2b+1$. Find the value when $b=-4$.

17. A diver’s position is represented by $d-5$. Find the value when $d=-2$.

18. The expression for a point score is $2x+y$. Find the score when $x=-3$ and $y=8$.

Potential Misunderstandings

• Students may think substitution means adding the given number to the expression instead of replacing the pronumeral
• Students may forget to replace every occurrence of the pronumeral in an expression
• Students may not recognise that $3x$ means multiplication
• Students may treat a negative substituted value as if the negative sign disappears
• Students may fail to use brackets when substituting a negative number and then make sign errors
• Students may ignore order of operations after substitution
• Students may confuse the operation sign in the expression with the sign of the substituted integer
• Students may make arithmetic errors when evaluating expressions with negative integers

Next: 055. Cartesian Plane