Linear Equations

17 Grade 9 Mathematics questions on Linear Equations, with worked marking schemes. 17 questions are available on this topic in full.

1. Question 1

   3 marks

   A line has gradient −2 and passes through (0, 5). Write its equation and find the x-intercept.

   (3 marks)

   Marking scheme

   A straight line is $y=mx+c$, where $m$ is the gradient and $c$ the $y$-intercept. The point $(0,5)$ is on the $y$-axis, so $c=5$. (1)

   With $m=-2$, the equation is $y=-2x+5$. (1)

   The line crosses the $x$-axis where $y=0$: $0=-2x+5$, so $2x=5$ and $x=2.5$. (1)

   Final answer: $y=-2x+5$, with $x$-intercept $2.5$.

2. Question 2

   3 marks

   Solve: 3(x + 2) = 2(x − 1) + 7

   (3 marks)

   Marking scheme

   Expand each bracket: the left becomes $3x+6$ and the right becomes $2x-2+7$. (1)

   Simplify the right: $-2+7=5$, so $3x+6=2x+5$. (1)

   Collect the $x$ terms on one side and the numbers on the other: $3x-2x=5-6$, giving $x=-1$. (1)

   Final answer: $x=-1$. Check: $3(-1+2)=3$ and $2(-1-1)+7=3$.

3. Question 3

   4 marks

   Solve simultaneously: 2x + y = 8 and x − y = 1.

   (4 marks)

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4. Question 4

   1 mark

   Solve: 4x − 3 = 2x + 9

   A. −6

   B. 12

   C. 6

   D. 3

   (1 mark)

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5. Question 5

   1 mark

   Solve for $x$: $3x-7=11$.

   A. $x=4$

   B. $x=18$

   C. $x=6$

   D. $x=1\frac{1}{3}$

   (1 mark)

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6. Question 6

   1 mark

   The gradient of the line joining (1, 2) and (3, 8) is ___

   A. 3

   B. 2

   C. 4

   D. 6

   (1 mark)

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7. Question 7

   1 mark

   The gradient of a horizontal line is ___

   A. 0

   B. ∞

   C. 1

   D. undefined

   (1 mark)

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8. Question 8

   1 mark

   Which point lies on the line y = 2x + 1?

   A. (1, 4)

   B. (3, 7)

   C. (0, 2)

   D. (2, 4)

   (1 mark)

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9. Question 9

   1 mark

   The y-intercept of y = 2x + 7 is ___

   A. 14

   B. −7

   C. 7

   D. 2

   (1 mark)

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10. Question 10

    4 marks

    Find the equation of the straight line passing through (2, 3) and (4, 7).

    (4 marks)

    Marking scheme

    The gradient between two points is $m=\frac{y_2-y_1}{x_2-x_1}$. (1)

    Substitute $(2,3)$ and $(4,7)$: $m=\frac{7-3}{4-2}=\frac{4}{2}=2$. (1)

    Use $y-y_1=m(x-x_1)$ with the point $(2,3)$: $y-3=2(x-2)$. (1)

    Expand and simplify: $y-3=2x-4$, so $y=2x-1$. Check $(4,7)$: $2(4)-1=7$. (1)

    Final answer: $y=2x-1$.

11. Question 11

    1 mark

    Find the equation of a line with gradient 2 passing through (0, −3).

    A. y = −2x − 3

    B. y = 2x − 3

    C. y = 2x + 3

    D. y = −3x + 2

    (1 mark)

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12. Question 12

    1 mark

    What is the gradient of the line y = 3x − 5?

    A. −5

    B. 5

    C. −3

    D. 3

    (1 mark)

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13. Question 13

    1 mark

    Solve: 2x + 5 = 13

    A. 9

    B. 3

    C. 8

    D. 4

    (1 mark)

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14. Question 14

    1 mark

    Solve the simultaneous equations: x + y = 7 and x − y = 3.

    A. x=4, y=3

    B. x=2, y=5

    C. x=3, y=4

    D. x=5, y=2

    (1 mark)

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15. Question 15

    1 mark

    Solve: 3x = 21

    A. 24

    B. 18

    C. 7

    D. 63

    (1 mark)

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16. Question 16

    3 marks

    The cost C of hiring a plumber is given by C = 500 + 200h, where h is the number of hours. How many hours were worked if the total cost was Ksh 1,700?

    (3 marks)

    Marking scheme

    Substitute the total cost into $C=500+200h$: $1,700=500+200h$. (1)

    The $500$ is the fixed charge, so subtract it: $200h=1,700-500=1,200$. (1)

    Each hour costs $200$, so divide: $h=\frac{1200}{200}=6$. (1)

    Final answer: $6$ hours.

17. Question 17

    3 marks

    At a shop in Kisumu, the cost of $5\text{kg}$ of maize flour together with a fixed transport charge of $\text{Ksh }30$ comes to $\text{Ksh }405$. Taking the cost of $1\text{kg}$ of flour as $x$, form an equation and find the cost per kilogram.

    (3 marks)

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