POP

G3.13 Algebraic Substitution

Image · 914 × 270

01 — Explanation

What does "substitution" mean?

Substitution means replacing each letter in an expression or formula with a given number, then working out the result using BIDMAS. $3x-5y$ with $x=-3,y=2$ means calculate $3\times(-3) - 5\times2$ , not " $3$ add $-3$ ."

02 — Explanation

How do I avoid sign errors when substituting negative numbers?

Always put the substituted value in brackets, every time — even for expressions that look simple.

5z^2 \text{ with } z=-10 \;\longrightarrow\; 5\times(-10)^2 = 5\times100=500

Common Misconception: Writing $5\times-10^2$ without brackets. Without them, BIDMAS squares the $10$ first and keeps the minus sign outside, giving the wrong answer $5\times(-100)=-500$ . The bracket is what protects the negative sign through the squaring.

03 — Explanation

How do I substitute into an expression with multiple terms and different operations?

Substitute every letter first, in brackets, before touching BIDMAS — this separates the two skills (translation, then calculation) so you're never trying to do both at once.

Example: Evaluate $4a-\dfrac{b}{2}+c^2$ when $a=3$ , $b=-8$ , $c=-2$ .

Substitute: $4(3)-\dfrac{(-8)}{2}+(-2)^2$ .
Work left to right through BIDMAS: $(-2)^2=4$ first (indices), then $4(3)=12$ and $(-8)\div2=-4$ .
Combine: $12-(-4)+4 = 12+4+4=20$ .

Exam Tip: When a fraction bar covers an expression like $\dfrac{b}{2}$ , treat the whole top as already "bracketed" — divide only after any arithmetic above the line is finished. The same logic applies to a fraction with an expression on both top and bottom, such as $\dfrac{a+b}{c}$ : simplify the numerator fully first, then divide by $c$ last.

04 — Explanation

What's the standard, mark-securing method?

Record the original expression/formula.
Substitute, bracketing every value.
Prioritize — apply BIDMAS (indices/powers before multiplication, multiplication before addition/subtraction).
Calculate the final number.

Exam Tip: Writing out the substitution line explicitly (step 2) earns you a method mark even if you go on to make an arithmetic slip — never skip straight to the answer.

05 — Explanation

How do I substitute into a formula and then solve for another letter?

Substitute first, then treat the result as a normal equation to solve.

Example: $P=2l+2w$ . Given $P=20$ cm, $w=4$ cm, find $l$ .

Substitute: $20 = 2l + 2(4)$ .
Simplify: $20=2l+8$ .
Solve: $2l=12$ , so $l=6$ cm.

06 — Explanation

How do I substitute into an expression involving a square root?

Work out everything inside the root first (following BIDMAS exactly as normal), and only take the square root as the very last step.

Example: Pythagoras' theorem states $c=\sqrt{a^2+b^2}$ . Find $c$ when $a=5$ and $b=12$ .

Substitute: $c=\sqrt{(5)^2+(12)^2}$ .
Square first: $(5)^2=25$ , $(12)^2=144$ .
Add inside the root: $25+144=169$ .
Take the square root last: $c=\sqrt{169}=13$ .

Common Misconception: Taking the square root of $a$ and $b$ separately before adding (e.g. $\sqrt{25}+\sqrt{144}=5+12=17$ ). The root applies to the whole sum, not to each term individually — always finish everything inside the root first.

07 — Worked example

Worked Example

$T = 2x^2 - 3y$ . Find the value of $T$ when $x=-4$ and $y=-5$ .

Record: $T=2x^2-3y$ .
Substitute, bracketing both values: $T=2(-4)^2-3(-5)$ .
Prioritize BIDMAS — square first: $(-4)^2=16$ . Then multiply: $2\times16=32$ and $3\times(-5)=-15$ .
Calculate: $T = 32 - (-15) = 32+15=47$ .