ALGEBRAIC OPERATIONS

QUESTION 1 [3 marks] — Criterion A Medium

Simplify each expression by collecting like terms:

a. $5x + 3y - 2x + 7y$

[1]

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$$3x+10y$$

b. $8a - 3b - 5a + 9b - 2a$

[1]

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$$a+6b$$

c. $4p^2 + 3p - p^2 - 5p + 6$

[1]

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$$3p^2-2p+6$$

QUESTION 2 [4 marks] — Criterion A Medium

Simplify: $7x - 4y + 3x^2 - 2x + 5y - x^2$

a. Group the like terms together (constants, $x$ terms, $y$ terms, $x^2$ terms).

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$x^2$ terms: $3x^2-x^2$. $x$ terms: $7x-2x$. $y$ terms: $-4y+5y$.

b. Simplify fully.

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$$2x^2+5x+y$$

QUESTION 3 [5 marks] — Criterion B Medium

Investigate whether simplifying an expression changes its value when a specific number is substituted.

a. Evaluate $5x+3-2x+7$ (unsimplified) at $x=4$.

[2]

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$5(4)+3-2(4)+7=20+3-8+7=22$

b. Simplify $5x+3-2x+7$, then evaluate the simplified version at $x=4$.

[2]

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Simplified: $3x+10$. At $x=4$: $3(4)+10=22$.

c. Compare your two answers, and explain why this makes sense.

[1]

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Both give 22 — simplifying doesn't change the VALUE of an expression, only how it's written, since collecting like terms is just applying the distributive law in reverse.

QUESTION 4 [4 marks] — Criterion C Medium

A student simplifies $3x + 2y$ as $5xy$.

a. Explain why this is incorrect, referring to what 'like terms' means.

[2]

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Only LIKE terms (terms with exactly the same variable part) can be combined by addition. $3x$ and $2y$ have different variables ($x$ vs $y$), so they are NOT like terms and cannot be added together into a single term — $3x+2y$ is already in simplest form.

b. Verify the student's error by evaluating both $3x+2y$ and $5xy$ at $x=2, y=3$.

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$3x+2y=3(2)+2(3)=6+6=12$. $5xy=5(2)(3)=30$. These are very different (12 vs 30), confirming $3x+2y\ne5xy$.

QUESTION 5 [4 marks] — Criterion D Medium

A gardener buys $x$ packets of tomato seeds at \$3 each and $y$ packets of lettuce seeds at \$2 each. On a second visit, they buy 2 more packets of tomato seeds and 3 more packets of lettuce seeds, at the same prices.

a. Write an expression for the total cost of the FIRST visit.

[1]

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$$3x+2y$$

b. Write an expression for the total cost of the SECOND visit.

[1]

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$$3(x+2)+2(y+3)$$

c. Write and simplify an expression for the TOTAL cost across both visits.

[2]

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$$3x+2y+3(x+2)+2(y+3) = 3x+2y+3x+6+2y+6 = 6x+4y+12$$

QUESTION 6 [5 marks] — Criterion A Hard

Simplify each expression fully:

a. $5x+3y-2x+7y-4$

[2]

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$$3x+10y-4$$

b. $3(2x-y)+2(x+3y)$

[3]

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$$6x-3y+2x+6y = 8x+3y$$

QUESTION 7 [5 marks] — Criterion A Hard

Simplify $3x^2-2x+5-x^2+4x-8$, then evaluate the simplified expression at $x=3$.

a. Simplify by collecting like terms.

[2]

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$$2x^2+2x-3$$

b. Evaluate at $x=3$, and verify by ALSO substituting $x=3$ into the ORIGINAL (unsimplified) expression, confirming both give the same result.

[3]

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Simplified: $2(9)+2(3)-3=18+6-3=21$. Original: $3(9)-2(3)+5-9+4(3)-8=27-6+5-9+12-8=21$ — matches, confirming the simplification was correct.

QUESTION 8 [5 marks] — Criterion B Hard

Investigate whether the ORDER in which you collect like terms (e.g. $x$ terms first vs constant terms first) ever affects the final simplified result.

a. Simplify $7x+4-3x+9-2x$ by collecting the $x$ terms FIRST, then the constants.

[2]

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$x$ terms: $7x-3x-2x=2x$. Constants: $4+9=13$. Result: $2x+13$.

b. Now simplify the SAME expression by collecting the constants FIRST, then the $x$ terms. Confirm you reach the same final answer, and explain why the order of collection never matters (referencing the commutative property of addition).

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Constants first: $4+9=13$. $x$ terms: $7x-3x-2x=2x$. Result: $2x+13$ — identical. This is guaranteed by the COMMUTATIVE property of addition: since addition can be performed in any order without changing the total, grouping and combining like terms in any sequence always produces the same final simplified expression.

QUESTION 9 [5 marks] — Criterion C Hard

A student simplifies $4x+3x^2$ as $7x^2$ (adding the coefficients and combining the different powers of $x$ into one term).

a. Explain precisely why $4x$ and $3x^2$ are NOT like terms, and therefore cannot be combined into a single term.

[2]

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Like terms must have the EXACT same variable part, including the same POWER of the variable. $4x$ has $x$ to the power 1, while $3x^2$ has $x$ to the power 2 — these are fundamentally different quantities (like trying to add a length to an area), so they cannot be combined; $4x+3x^2$ is already in its simplest form.

b. Verify the error by evaluating both $4x+3x^2$ and the student's incorrect $7x^2$ at $x=2$.

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$4x+3x^2$ at $x=2$: $8+12=20$. Student's $7x^2$ at $x=2$: $7(4)=28$. These are different ($20\ne28$), confirming the terms cannot be combined.

QUESTION 10 [4 marks] — Criterion C Hard

A classmate says: 'like terms must have the exact same NUMBER of variables multiplied together, so $xy$ and $x^2y$ can be combined since they both have $x$ and $y$ in them.'

a. Explain why $xy$ and $x^2y$ are actually NOT like terms, correcting the classmate's misunderstanding about what makes terms 'alike'.

[2]

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Like terms require the EXACT same combination of variables raised to the EXACT same powers — $xy$ means $x^1y^1$, while $x^2y$ means $x^2y^1$. Even though both terms involve $x$ and $y$, the DIFFERENT power of $x$ (1 vs 2) makes them fundamentally different terms, just like $4x$ and $3x^2$ from before.

b. Give ONE example of a term that WOULD be a genuine like term with $xy$ (i.e. could be legitimately combined with it), and explain why it qualifies.

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Example: $5xy$ (or any multiple of $xy$, such as $-2xy$ or $\frac{1}{2}xy$) — it qualifies because it has the exact same variable parts with the exact same powers ($x^1y^1$), differing only in the numerical coefficient, which is precisely what makes terms 'like' and combinable.

QUESTION 11 [4 marks] — Criterion D Hard

A landscaper charges a callout fee of \$50, plus \$15 per hour of labour ($h$ hours), plus \$8 per bag of mulch ($b$ bags). A customer orders TWO separate jobs on the same day: Job 1 needs 3 hours and 5 bags; Job 2 needs 2 hours and 3 bags.

a. Write a simplified SINGLE expression for the TOTAL cost of both jobs combined, in terms of a single callout fee (charged once) plus combined labour and mulch costs.

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Total labour hours: $3+2=5$. Total mulch bags: $5+3=8$. Total cost (one callout fee): $$50+15(5)+8(8)=50+75+64=189$$

Collecting like terms