Simultaneous Linear Equations (Grade 10) — Worksheet & Memo

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Grade 10 · Equations & Inequalities

Simultaneous Linear Equations (Grade 10)

MARKING GUIDELINE

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 36

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
Solve simultaneously: x + y = 10 and x − y = 4.
(1)
A)x = 7 and y = 3✓
B)x = 3 and y = 7
C)x = 6 and y = 4
D)x = 14 and y = 6
Answer: A — Adding the two equations gives 2x = 14, so x = 7 and then y = 3.
- B — answered x and y the wrong way round
- C — used 10 − 4 = 6 for x without halving the sum
- D — added the equations but never divided the 14 by 2
1.2
Solve simultaneously: y = 2x + 1 and 3x + y = 11.
(1)
A)x = 2 and y = 5✓
B)x = 2 and y = 4
C)x = 5 and y = 2
D)x = 3 and y = 7
Answer: A — Substituting gives 3x + (2x + 1) = 11, so 5x = 10, x = 2 and y = 5.
- B — substituted x = 2 into 2x and forgot the + 1
- C — answered x and y the wrong way round
- D — guessed at 3x + y = 11 alone, ignoring the first equation
1.3
Solve simultaneously: 2x + 3y = 12 and 2x − y = 4.
(1)
A)x = 3 and y = 2✓
B)x = 3 and y = 4
C)x = 2 and y = 3
D)x = 6 and y = 0
Answer: A — Subtracting the second from the first gives 4y = 8, so y = 2 and then 2x = 6.
- B — added the equations instead of subtracting, so the 2x never cancelled
- C — answered x and y the wrong way round
- D — solved only the second equation and ignored the first
1.4
On a set of axes, the solution of a pair of simultaneous linear equations is …
(1)
A)the point where the two lines cut✓
B)the y-intercept of both lines
C)the gradient of both lines
D)the point where each line cuts the x-axis
Answer: A — The solution is the ONE pair (x ; y) satisfying both equations, which is where the graphs meet.
- B — the y-intercept is where a line meets the y-axis, not where the lines meet each other
- C — the gradient is a rate of change, not a solution
- D — each x-intercept lies on ONE line only
1.5
If the two lines are PARALLEL, the simultaneous equations have …
(1)
A)no solution✓
B)one solution
C)two solutions
D)infinitely many solutions
Answer: A — Parallel lines never meet, so no pair (x ; y) can satisfy both equations.
- B — one solution needs the lines to CUT, which parallel lines never do
- C — two straight lines can never cut each other twice
- D — that happens when the two equations describe the SAME line, not parallel ones
1.6
Three pens and two books cost R74. One pen and one book cost R32. A book costs …
(1)
A)R22✓
B)R10
C)R32
D)R42
Answer: A — p + b = 32 gives p = 32 − b; substituting into 3p + 2b = 74 leaves b = 22.
- B — that is the price of a PEN, not of a book
- C — that is the cost of a pen AND a book together
- D — subtracted the two equations without first matching the coefficients
1.7
Solve simultaneously: 5x + 2y = 16 and 3x − 2y = 8.
(1)
A)x = 3 and y = 0,5✓
B)x = 3 and y = 1
C)x = 3 and y = −0,5
D)x = 24 and y = 0,5
Answer: A — Adding eliminates y: 8x = 24, so x = 3, and 15 + 2y = 16 gives y = 0,5.
- B — solved 2y = 1 as y = 1 instead of halving
- C — dropped the minus on the 2y when substituting back
- D — added the equations but never divided the 24 by 8
1.8
The sum of two numbers is 25 and their difference is 7. The SMALLER number is …
(1)
A)9✓
B)16
C)18
D)12,5
Answer: A — Adding gives 2x = 32, so the larger number is 16 and the smaller is 25 − 16 = 9.
- B — that is the LARGER number
- C — used 25 − 7 = 18 without halving
- D — halved 25 and ignored the difference altogether
1.9
To eliminate y from 3x + 2y = 7 and 5x + 3y = 11, multiply …
(1)
A)the first by 3 and the second by 2✓
B)the first by 5 and the second by 3
C)the first by 2 and the second by 3
D)both equations by 6
Answer: A — 2y × 3 and 3y × 2 both become 6y, so subtracting removes y.
- B — those multipliers match the x coefficients, so they eliminate x, not y
- C — that gives 4y and 9y, which do not cancel
- D — multiplying BOTH by 6 changes nothing — the coefficients still differ
1.10
Is (x ; y) = (4 ; −1) a solution of 2x + 3y = 5 and x − y = 5?
(1)
A)yes — it satisfies both equations✓
B)no — it satisfies neither
C)it satisfies only the first
D)it satisfies only the second
Answer: A — 2(4) + 3(−1) = 8 − 3 = 5, and 4 − (−1) = 5.
- B — both checks do work — substitute and see
- C — 4 − (−1) = 5 as well, so the second holds too
- D — 8 − 3 = 5 as well, so the first holds too

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[12 MARKS]

Solve the following simultaneously. Show ALL your working.

2.1
2x + 4k = 20 and 3x − 2k = 6
(5)
Multiply the second equation by 2: 6x − 4k = 12 (1)
Add: 8x = 32 (2)
x = 4 (1)
k = 3 (1)
2.2
y = 2x and x + y = 9
(3)
x + 2x = 9 (1)
3x = 9 (1)
x = 3 and y = 6 (1)
2.3
The sum of three consecutive integers is 39. Determine, WITH THE USE OF AN EQUATION, the three numbers.
(4)
Let them be x, x + 1 and x + 2 (1)
3x + 3 = 39 (1)
x = 12 (1)
12, 13 and 14 (1)

Question 3

[14 MARKS]

Answer the word problems below, using simultaneous equations.

3.1
Three apples and two pears cost R16, while one apple and four pears cost R12. Determine the price of each fruit.
(5)
3a + 2p = 16 and a + 4p = 12 (2)
From the second: a = 12 − 4p (1)
3(12 − 4p) + 2p = 16 gives p = 2 (1)
an apple R4, a pear R2 (1)
3.2
Explain what the solution of a pair of simultaneous linear equations represents graphically, and what it means if there is NO solution.
(4)
The solution is the point where the two lines INTERSECT (2)
No solution means the lines are PARALLEL (1) and never meet (1)
3.3
A cinema charges R60 for an adult and R35 for a child. 120 tickets were sold for R5 725. Determine how many of each were sold.
(5)
a + c = 120 and 60a + 35c = 5 725 (2)
c = 120 − a (1)
60a + 35(120 − a) = 5 725, so 25a = 1 525 (1)
61 adults and 59 children (1)

TOTAL: 36 marks

This question paper consists of 3 questions.