Linear Equations

15 Grade 7 Mathematics questions on Linear Equations, with worked marking schemes. 15 questions are available on this topic in full.

1. Question 1

   4 marks

   Solve simultaneously:

   (4 marks)

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2. Question 2

   1 mark

   If 5x + 7 = 3x + 19, find x.

   A. 8

   B. 3

   C. 6

   D. 12

   (1 mark)

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3. Question 3

   3 marks

   Solve the equation:

   $$3x+7=22$$

   (3 marks)

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4. Question 4

   1 mark

   Solve: $\frac{x}{3}+5=9$

   A. 42

   B. 4

   C. 12

   D. 2

   (1 mark)

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5. Question 5

   4 marks

   Solve: $\frac{x-2}{4}=\frac{x+1}{6}$

   (4 marks)

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   Clear the fractions by cross-multiplying: $6(x-2)=4(x+1)$. (1)

   Expand both brackets: $6x-12=4x+4$. (1)

   Collect like terms: $6x-4x=4+12$, so $2x=16$. (1)

   Divide by $2$: $x=8$. Check: $\frac{8-2}{4}=\frac{3}{2}$ and $\frac{8+1}{6}=\frac{3}{2}$. (1)

   Final answer: $x=8$.

6. Question 6

   4 marks

   Ouma is twice as old as Atieno. In 5 years, the sum of their ages will be 40. Find their present ages.

   (4 marks)

   Marking scheme

   Let Atieno be $x$ years old. Ouma is twice as old, so Ouma is $2x$. (1)

   In $5$ years each is $5$ years older: $(x+5)+(2x+5)=40$. (1)

   Simplify: $3x+10=40$, so $3x=30$. (1)

   Divide by $3$: $x=10$, so Ouma is $2\times 10=20$. Check: $15+25=40$. (1)

   Final answer: Atieno is $10$ and Ouma is $20$.

7. Question 7

   1 mark

   Solve: $\frac{2x+1}{3}=5$

   A. 8

   B. 7

   C. 2

   D. 14

   (1 mark)

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8. Question 8

   1 mark

   Solve: $4x-3=13$

   A. 16

   B. 4

   C. 10

   D. 2.5

   (1 mark)

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9. Question 9

   1 mark

   The equation of a line is $4x-3y-7=0$. What is the gradient?

   A. 4

   B. $\frac{3}{4}$

   C. $\frac{4}{3}$

   D. 3

   (1 mark)

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10. Question 10

    2 marks

    The equation of a line was given as $4x-3y-7=0$. What is the gradient of the line?

    A. 3

    B. ³⁄₄

    C. ⁴⁄₃

    D. 4

    Marking scheme

    C. $\frac{4}{3}$ — rearrange into $y=mx+c$, where $m$ is the gradient: $3y=4x-7$, so $y=\frac{4}{3}x-\frac{7}{3}$. The option $\frac{3}{4}$ turns the fraction upside down, and reading a coefficient straight off the unrearranged equation gives $4$ or $3$.

11. Question 11

    4 marks

    The sum of three consecutive even numbers is 78. Find the numbers.

    (4 marks)

    Marking scheme

    Consecutive even numbers rise in steps of $2$, so call them $n$, $n+2$ and $n+4$. (1)

    Their sum is $n+(n+2)+(n+4)=3n+6$, and this equals $78$. (1)

    Subtract $6$: $3n=72$, so $n=24$. (1)

    The numbers are $24$, $26$ and $28$. Check: $24+26+28=78$. (1)

    Final answer: $24$, $26$ and $28$.

12. Question 12

    3 marks

    Solve: $3x+7=22$

    (3 marks)

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13. Question 13

    2 marks

    The equation of a line was given as $4x-3y-7=0$. What is the gradient of the line?

    A. 3

    B. ³⁄₄

    C. ⁴⁄₃

    D. 4

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14. Question 14

    3 marks

    Solve the equation:

    $$3x+7=22$$

    (3 marks)

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15. Question 15

    1 mark

    Solve: $3(2x-1)=15$

    A. 6

    B. 3

    C. 5

    D. 2

    (1 mark)

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