034. Substituting into Algebraic Expressions

Learning Intentions

• To understand that variables/pronumerals can be replaced with numbers
• substitute numbers for variables/pronumerals
• evaluate an expression using order of operations once all variable/pronumeral values are known

Pre-requisite Summary

• Understand that a variable or pronumeral is a symbol that represents a number
• Know that substitution means replacing a variable with its given value
• Recognise that multiplication can be written without a sign, for example $3x=3\times x$
• Recall the order of operations for expressions involving more than one operation
• Be able to use brackets correctly when substituting negative or multi-step values if needed
• Be able to perform basic arithmetic accurately after substitution

Worked Examples

Worked Example 1

Substitute the given value and evaluate:
a) Find the value of $x+5$ when $x=3$
b) Find the value of $2a$ when $a=7$
c) Explain what substitution means

Worked Example 2

Substitute and evaluate:
a) Find the value of $3x+4$ when $x=5$
b) Find the value of $2y-6$ when $y=9$

Worked Example 3

Substitute and evaluate using order of operations:
a) Find the value of $4m+2n$ when $m=3$ and $n=5$
b) Find the value of $6p-2q$ when $p=4$ and $q=7$

Worked Example 4

Substitute and evaluate using order of operations:
a) Find the value of $3x+2\times y$ when $x=4$ and $y=6$
b) Find the value of $12-2a+b$ when $a=3$ and $b=5$

Worked Example 5

Substitute and evaluate expressions with brackets:
a) Find the value of $2(x+3)$ when $x=4$
b) Find the value of $5(a-1)$ when $a=6$

Worked Example 6

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

Problems

Problem 1

Substitute the given value and evaluate:
a) Find the value of $x+7$ when $x=2$
b) Find the value of $3a$ when $a=6$
c) Explain what substitution means

Problem 2

Substitute and evaluate:
a) Find the value of $4x+1$ when $x=3$
b) Find the value of $5y-2$ when $y=8$

Problem 3

Substitute and evaluate using order of operations:
a) Find the value of $2m+3n$ when $m=4$ and $n=2$
b) Find the value of $7p-q$ when $p=5$ and $q=6$

Problem 4

Substitute and evaluate using order of operations:
a) Find the value of $2x+3\times y$ when $x=5$ and $y=4$
b) Find the value of $15-3a+b$ when $a=2$ and $b=7$

Problem 5

Substitute and evaluate expressions with brackets:
a) Find the value of $3(x+2)$ when $x=5$
b) Find the value of $4(a-2)$ when $a=7$

Problem 6

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

Problems

Problem 1

a) Evaluate $x+6$ when $x=4$
b) Evaluate $2a$ when $a=9$

Problem 2

a) Evaluate $5x+2$ when $x=3$
b) Evaluate $4y-7$ when $y=6$

Problem 3

a) Evaluate $3m+2n$ when $m=2$ and $n=5$
b) Evaluate $8p-3q$ when $p=4$ and $q=2$

Problem 4

a) Evaluate $2x+4\times y$ when $x=3$ and $y=5$
b) Evaluate $20-2a+b$ when $a=6$ and $b=4$

Problem 5

a) Evaluate $4(x+1)$ when $x=6$
b) Evaluate $3(a-2)$ when $a=8$

Problem 6

a) Evaluate $x^2+4x$ when $x=3$
b) Evaluate $2a+b^2$ when $a=6$ and $b=4$

Exercises

Understanding and Fluency

1. Substitute and evaluate:
   a) $x+4$ when $x=2$
   b) $a+9$ when $a=5$
   c) $b+7$ when $b=1$

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

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

4. Substitute and evaluate:
   a) $5x-3$ when $x=6$
   b) $2y+8$ when $y=5$
   c) $7n-4$ when $n=3$

5. Substitute and evaluate using two variables:
   a) $x+y$ when $x=3,y=8$
   b) $2a+b$ when $a=4,b=5$
   c) $3m+2n$ when $m=2,n=6$

6. Substitute and evaluate using two variables:
   a) $4p-q$ when $p=5,q=7$
   b) $2r+3s$ when $r=1,s=4$
   c) $6u-2v$ when $u=3,v=5$

7. Substitute and evaluate using order of operations:
   a) $3x+2\times y$ when $x=2,y=5$
   b) $12-2a+b$ when $a=4,b=3$
   c) $4m+3n-2$ when $m=1,n=6$

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

9. Substitute and evaluate powers:
   a) $x^2$ when $x=6$
   b) $a^2+3$ when $a=4$
   c) $b^2+2b$ when $b=5$

10. Mixed substitution:
    a) $2x+y$ when $x=4,y=3$
    b) $3a+2b$ when $a=2,b=6$
    c) $p^2+q$ when $p=3,q=8$

Reasoning

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

12. A student says that if $x=4$, then $2x+5=245$. Explain the mistake.

13. Explain why order of operations is still needed after substituting values into an expression.

14. A student substitutes $x=3$ into $x^2+2$ and gets $3^2+2=3\times 2+2=8$. Explain why this is incorrect.

Problem-solving

15. A taxi fare is given by the expression $4k+6$, where $k$ is the number of kilometres travelled. Find the fare when $k=5$.

16. The total cost of buying $n$ notebooks at $3 each and one folder costing $4 is given by $3n+4$. Find the cost when $n=7$.

17. The perimeter of a rectangle with length $x+2$ and width $3$ is given by $2(x+2)+2\times 3$. Find the perimeter when $x=5$.

18. A phone plan costs $10 plus $2 for each gigabyte $g$ used. The total cost is $10+2g$. Find the cost when $g=8$.

Potential Misunderstandings

• Students may think substitution means adding the variable value to the expression without replacing the variable first
• Students may not recognise that $3x$ means $3\times x$
• Students may forget to replace every occurrence of the variable in an expression
• Students may ignore brackets when substituting and evaluating
• Students may not apply order of operations after substitution
• Students may confuse $x^2$ with $2x$
• Students may treat adjacent numbers as separate digits, for example reading $2x$ with $x=4$ as $24$ instead of $2\times 4$
• Students may substitute correctly but make arithmetic errors in the evaluation step

Next: 035. Equivalent Expressions and Algebraic Generalisation