Algebra

Simultaneous equations

Solve two linear equations by substitution or elimination. Graphically: the intersection of two lines.

CAPS

Grade 10 · Mathematics · Algebra

This sits on Mathematics Paper 1.

The paper
Paper 1 is algebra, exponents, equations, number patterns, functions, finance and probability. Copy the formula, then substitute.
Cognitive level
CAPS spreads the marks across knowledge, routine procedures, complex procedures and problem solving. A routine question still wants the line of working. A problem-solving question wants you to choose the method, not only to calculate.
The relation
ax + by = c. Standard linear form
How a marker reads it
Gauteng Paper 1 wants the pair written as x = …, y = …, not only a point (x; y), unless they ask for coordinates.

Each Mathematics paper is 150 marks. A correct substitution can earn a method mark even when the last arithmetic slips, if the formula was the right one.

Diagram

Worked example

Solve: 2x + y = 11 and x − y = 1.

Before any working — hunt first

Given

Numbers and units already in the sentence: 2x · 11 and · 1. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: Solve: 2x + y = 11 and x − y = 1. Circle that word — that is the missing variable.

Find

We are required to find: Solve: 2x + y = 11 and x − y = 1. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write ax + by = c with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Solve: 2x + y = 11 and x − y = 1.

Simplified approach

Add the two equations to kill y, then back-substitute.

Culture note

Two taxi fares from a Johannesburg rank — one cash, one change — two facts, one pair of numbers that fits both.

Formulas

ax+by=c

Standard linear form

How to hunt the missing letter

Do not start by naming the letters. First hunt the missing one: copy every number you can see (Given). The letter with no number is Missing. The asking word is Find. The formula that still contains that letter is what you Calculate.

How to hunt: (1) Underline every number and its unit — those are Given. (2) Circle the asking word — find, calculate, how far, what is. That word names Find. (3) The letter the asking word names, which has no number yet, is Missing. (4) Write the formula with empty letters. Cross out every given letter. The letter still standing is what you Calculate.

x
first unknown
How you spot this in the sentence
When the sentence names the first unknown, that letter is x. Copy the number next to it into Given. If the sentence never names it, x is Missing. Usually the first letter in ax + by = c.
Example:In x + y = 5 and x − y = 1, x = 3.
y
second unknown
How you spot this in the sentence
When the sentence names the second unknown, that letter is y. Copy the number next to it into Given. If the sentence never names it, y is Missing. The other letter. Eliminate one unknown, then back-substitute.
Example:Once x=3, y=5−3=2.
a, b, c
coefficients
How you spot this in the sentence
When the sentence names the coefficients, that letter is a, b, c. Copy the number next to it into Given. If the sentence never names it, a, b, c is Missing. a is in front of x, b in front of y, c is the constant.
Example:2x+3y=12 has a=2, b=3, c=12.

Exam tricks you must know

These are the moves markers look for — not shortcuts that skip the method, but the habits that earn the method mark.

“Factorise fully” means HCF and then the identity — leaving 2(x² − 9) loses a mark
How:Look at the given polynomial. Pull out the biggest number and letter that divides every term (HCF). Then look at what is left: difference of squares, trinomial, or grouping. Both layers go on the page. Stop only when nothing else factorises.
When to use it:Any “factorise fully” on Paper 1, especially KZN.
Memorise exact values for 0°, 30°, 45°, 60°, 90° — compound angles collapse without them
How:Before a calculator, write the triangle: 30-60-90 and 45-45-90. sin 30° = 12, sin 45° = 1√2, sin 60° = √32, and the cosines swap. Compound-angle questions want these as fractions with division lines, not decimals.
When to use it:Reduction, compound angles, and “without a calculator” trig.
Name similar triangles in corresponding order: ΔABC ∼ ΔDEF, not ΔABC ∼ ΔEFD
How:Hunt equal angles first and mark them on the sketch. Then list vertices so that equal angles sit in the same position: A maps to D, B to E, C to F. The ratio statement uses that same order. Wrong order loses the reason mark even if the lengths are right.
When to use it:Similarity, intercept, and 3-D trig that shares a triangle.

Step 1 / 6

Add

(2x + y)+(x − y)=11+1→3x=12→x=4.

x=4

Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

4. Step-by-step working

01
Add
(2x + y)+(x − y)=11+1→3x=12→x=4.
x=4
02
Back-substitute
4−y=1→y=3.
y=3
03
Check
2(4)+3=11. Both equations hold.
04
Step F — Box the answer with its unit
x=4, y=3

More worked examples

Board walkthroughs on this topic. Open one, copy the working, then try the next.

Example 1 / 20

Solve: x + y = 5 and x − y = -1.

4. Step-by-step working

01
Add
(x+y)+(x−y)=5+(-1)→2x=4→x=2.
x=2
02
Substitute
2+y=5→y=3.
y=3

How we got there

Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.

Explain in another language · 11

1. What is the question asking?

Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Solve: x + y = 5 and x − y = -1.

Before any working — hunt first

Given

Numbers and units already in the sentence: 5 and · 1. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: Solve: x + y = 5 and x − y = -1. Circle that word — that is the missing variable.

Find

We are required to find: Solve: x + y = 5 and x − y = -1. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write x = 2 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Solve: x + y = 5 and x − y = -1.

In other words — as if you have never seen this before

If you have never seen this before: Add to kill y, then back-substitute. We will name a method in words, write the formula with no numbers, then substitute one line at a time.

2. What do we already know?

Eq1: x+y=5. Eq2: x−y=-1.

3. Which method — and why this one?

Add to kill y, then back-substitute.

4. Step-by-step working

01 / 05
Add
(x+y)+(x−y)=5+(-1)→2x=4→x=2.
x=2
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 05
Substitute
2+y=5→y=3.
y=3
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
x=2, y=3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.

5. The answer

x=2, y=3

Copy onto your answer sheet — in this order

Formula: x=2

Answer: x=2, y=3

6. Check it like a marker

Check eq2: 2−3=-1.

7. A trap first-timers fall into

First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”

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