Mathbench

3. Straight lines

The gradient of the line through two points, and the equation that follows from it. Set for OMPT-A, OMPT-B, OMPT-C, OMPT-D, OMPT-E, OMPT-F and OMPT-G.

All OMPT worksheets

Questions

Sheet length
1

Give the gradient of the line through A(7; −2)A(7;\ -2) and B(11; −4)B(11;\ -4).

Answer:

2

Solve for xx: 5(x−2)=2x−75\left(x - 2\right) = 2 x - 7

Answer:

3

Solve for xx: x+55=x+46\dfrac{x + 5}{5} = \dfrac{x + 4}{6}

Answer:

4

Give the gradient of the line through A(8; 4)A(8;\ 4) and B(6; 7)B(6;\ 7).

Answer:

5

Solve for xx: 6(x+8)=5x+36\left(x + 8\right) = 5 x + 3

Answer:

6

Solve for xx: x−64=x−36\dfrac{x - 6}{4} = \dfrac{x - 3}{6}

Answer:

7

Give the gradient of the line through A(7; −1)A(7;\ -1) and B(6; 2)B(6;\ 2).

Answer:

8

Solve for xx: 2(x+5)=6x+92\left(x + 5\right) = 6 x + 9

Answer:

9

Solve for xx: x−72=x+36\dfrac{x - 7}{2} = \dfrac{x + 3}{6}

Answer:

10

Give the gradient of the line through A(−8; −6)A(-8;\ -6) and B(−12; −5)B(-12;\ -5).

Answer:

11

Solve for xx: 7(x−4)=4x−57\left(x - 4\right) = 4 x - 5

Answer:

12

Solve for xx: x+32=x+54\dfrac{x + 3}{2} = \dfrac{x + 5}{4}

Answer:

13

Give the gradient of the line through A(2; 7)A(2;\ 7) and B(−1; −2)B(-1;\ -2).

Answer:

14

Solve for xx: 3(x−5)=5x+63\left(x - 5\right) = 5 x + 6

Answer:

15

Solve for xx: x+45=x−62\dfrac{x + 4}{5} = \dfrac{x - 6}{2}

Answer:

16

Give the gradient of the line through A(−5; −2)A(-5;\ -2) and B(−3; −9)B(-3;\ -9).

Answer:

17

Solve for xx: 5(x−5)=6x−35\left(x - 5\right) = 6 x - 3

Answer:

18

Solve for xx: x+43=x−96\dfrac{x + 4}{3} = \dfrac{x - 9}{6}

Answer:

19

Give the gradient of the line through A(−5; 2)A(-5;\ 2) and B(−2; 0)B(-2;\ 0).

Answer:

20

Solve for xx: 5(x−5)=6x−45\left(x - 5\right) = 6 x - 4

Answer:

21

Solve for xx: x−53=x−12\dfrac{x - 5}{3} = \dfrac{x - 1}{2}

Answer:

22

Give the gradient of the line through A(−1; −5)A(-1;\ -5) and B(0; −7)B(0;\ -7).

Answer:

23

Solve for xx: 7(x+6)=6x−47\left(x + 6\right) = 6 x - 4

Answer:

24

Solve for xx: x+92=x−46\dfrac{x + 9}{2} = \dfrac{x - 4}{6}

Answer:

25

Give the gradient of the line through A(5; 2)A(5;\ 2) and B(6; −6)B(6;\ -6).

Answer:

26

Solve for xx: 5(x+8)=3x+25\left(x + 8\right) = 3 x + 2

Answer:

27

Solve for xx: x+96=x−45\dfrac{x + 9}{6} = \dfrac{x - 4}{5}

Answer:

28

Give the gradient of the line through A(−4; 2)A(-4;\ 2) and B(−7; 7)B(-7;\ 7).

Answer:

29

Solve for xx: 2(x−5)=6x−82\left(x - 5\right) = 6 x - 8

Answer:

30

Solve for xx: x−94=x−33\dfrac{x - 9}{4} = \dfrac{x - 3}{3}

Answer:

Answers

Every answer below was re-derived independently before this page was built.

1

Answer: −12- \frac{1}{2}

m=y2−y1x2−x1=−4−(−2)11−(7)=−12m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-4 - (-2)}{11 - (7)} = - \frac{1}{2}.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −2-2
  • Subtracted the coordinates in opposite orders → 12\frac{1}{2}
2

Answer: 11

Expand the bracket: 5x−10=2x−75 x - 10 = 2 x - 7. Collect the xx terms on one side and the numbers on the other, then divide: x=1x = 1.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −53- \frac{5}{3}
  • Moved a term across without changing its sign → 88
3

Answer: −10-10

Cross-multiply to clear both denominators: 6(x+5)=5(x+4)6\left(x + 5\right) = 5\left(x + 4\right). Expand, collect the xx terms and divide: x=−10x = -10.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → −26-26
  • Added the fractions instead of cross-multiplying → −5011- \frac{50}{11}
4

Answer: −32- \frac{3}{2}

m=y2−y1x2−x1=7−(4)6−(8)=−32m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{7 - (4)}{6 - (8)} = - \frac{3}{2}.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −23- \frac{2}{3}
  • Subtracted the coordinates in opposite orders → 32\frac{3}{2}
5

Answer: −45-45

Expand the bracket: 6x+48=5x+36 x + 48 = 5 x + 3. Collect the xx terms on one side and the numbers on the other, then divide: x=−45x = -45.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −5-5
  • Moved a term across without changing its sign → −54-54
6

Answer: 1212

Cross-multiply to clear both denominators: 6(x−6)=4(x−3)6\left(x - 6\right) = 4\left(x - 3\right). Expand, collect the xx terms and divide: x=12x = 12.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → 332\frac{33}{2}
  • Added the fractions instead of cross-multiplying → 245\frac{24}{5}
7

Answer: −3-3

m=y2−y1x2−x1=2−(−1)6−(7)=−3m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{2 - (-1)}{6 - (7)} = -3.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −13- \frac{1}{3}
  • Subtracted the coordinates in opposite orders → 33
8

Answer: 14\frac{1}{4}

Expand the bracket: 2x+10=6x+92 x + 10 = 6 x + 9. Collect the xx terms on one side and the numbers on the other, then divide: x=14x = \frac{1}{4}.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −1-1
  • Moved a term across without changing its sign → 77
9

Answer: 1212

Cross-multiply to clear both denominators: 6(x−7)=2(x+3)6\left(x - 7\right) = 2\left(x + 3\right). Expand, collect the xx terms and divide: x=12x = 12.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → 454\frac{45}{4}
  • Added the fractions instead of cross-multiplying → 92\frac{9}{2}
10

Answer: −14- \frac{1}{4}

m=y2−y1x2−x1=−5−(−6)−12−(−8)=−14m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-5 - (-6)}{-12 - (-8)} = - \frac{1}{4}.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −4-4
  • Subtracted the coordinates in opposite orders → 14\frac{1}{4}
11

Answer: 233\frac{23}{3}

Expand the bracket: 7x−28=4x−57 x - 28 = 4 x - 5. Collect the xx terms on one side and the numbers on the other, then divide: x=233x = \frac{23}{3}.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −13- \frac{1}{3}
  • Moved a term across without changing its sign → 383\frac{38}{3}
12

Answer: −1-1

Cross-multiply to clear both denominators: 4(x+3)=2(x+5)4\left(x + 3\right) = 2\left(x + 5\right). Expand, collect the xx terms and divide: x=−1x = -1.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → −72- \frac{7}{2}
  • Added the fractions instead of cross-multiplying → −113- \frac{11}{3}
13

Answer: 33

m=y2−y1x2−x1=−2−(7)−1−(2)=3m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-2 - (7)}{-1 - (2)} = 3.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → 13\frac{1}{3}
  • Subtracted the coordinates in opposite orders → −3-3
14

Answer: −212- \frac{21}{2}

Expand the bracket: 3x−15=5x+63 x - 15 = 5 x + 6. Collect the xx terms on one side and the numbers on the other, then divide: x=−212x = - \frac{21}{2}.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −112- \frac{11}{2}
  • Moved a term across without changing its sign → −32- \frac{3}{2}
15

Answer: 383\frac{38}{3}

Cross-multiply to clear both denominators: 2(x+4)=5(x−6)2\left(x + 4\right) = 5\left(x - 6\right). Expand, collect the xx terms and divide: x=383x = \frac{38}{3}.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → 143\frac{14}{3}
  • Added the fractions instead of cross-multiplying → 227\frac{22}{7}
16

Answer: −72- \frac{7}{2}

m=y2−y1x2−x1=−9−(−2)−3−(−5)=−72m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-9 - (-2)}{-3 - (-5)} = - \frac{7}{2}.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −27- \frac{2}{7}
  • Subtracted the coordinates in opposite orders → 72\frac{7}{2}
17

Answer: −22-22

Expand the bracket: 5x−25=6x−35 x - 25 = 6 x - 3. Collect the xx terms on one side and the numbers on the other, then divide: x=−22x = -22.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −2-2
  • Moved a term across without changing its sign → −31-31
18

Answer: −17-17

Cross-multiply to clear both denominators: 6(x+4)=3(x−9)6\left(x + 4\right) = 3\left(x - 9\right). Expand, collect the xx terms and divide: x=−17x = -17.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → −11-11
  • Added the fractions instead of cross-multiplying → 13\frac{1}{3}
19

Answer: −23- \frac{2}{3}

m=y2−y1x2−x1=0−(2)−2−(−5)=−23m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{0 - (2)}{-2 - (-5)} = - \frac{2}{3}.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −32- \frac{3}{2}
  • Subtracted the coordinates in opposite orders → 23\frac{2}{3}
20

Answer: −21-21

Expand the bracket: 5x−25=6x−45 x - 25 = 6 x - 4. Collect the xx terms on one side and the numbers on the other, then divide: x=−21x = -21.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −1-1
  • Moved a term across without changing its sign → −33-33
21

Answer: −7-7

Cross-multiply to clear both denominators: 2(x−5)=3(x−1)2\left(x - 5\right) = 3\left(x - 1\right). Expand, collect the xx terms and divide: x=−7x = -7.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → −9-9
  • Added the fractions instead of cross-multiplying → 135\frac{13}{5}
22

Answer: −2-2

m=y2−y1x2−x1=−7−(−5)0−(−1)=−2m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-7 - (-5)}{0 - (-1)} = -2.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −12- \frac{1}{2}
  • Subtracted the coordinates in opposite orders → 22
23

Answer: −46-46

Expand the bracket: 7x+42=6x−47 x + 42 = 6 x - 4. Collect the xx terms on one side and the numbers on the other, then divide: x=−46x = -46.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −10-10
  • Moved a term across without changing its sign → −34-34
24

Answer: −312- \frac{31}{2}

Cross-multiply to clear both denominators: 6(x+9)=2(x−4)6\left(x + 9\right) = 2\left(x - 4\right). Expand, collect the xx terms and divide: x=−312x = - \frac{31}{2}.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → −292- \frac{29}{2}
  • Added the fractions instead of cross-multiplying → −234- \frac{23}{4}
25

Answer: −8-8

m=y2−y1x2−x1=−6−(2)6−(5)=−8m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-6 - (2)}{6 - (5)} = -8.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −18- \frac{1}{8}
  • Subtracted the coordinates in opposite orders → 88
26

Answer: −19-19

Expand the bracket: 5x+40=3x+25 x + 40 = 3 x + 2. Collect the xx terms on one side and the numbers on the other, then divide: x=−19x = -19.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → −3-3
  • Moved a term across without changing its sign → −22-22
27

Answer: 6969

Cross-multiply to clear both denominators: 5(x+9)=6(x−4)5\left(x + 9\right) = 6\left(x - 4\right). Expand, collect the xx terms and divide: x=69x = 69.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → 4949
  • Added the fractions instead of cross-multiplying → −2111- \frac{21}{11}
28

Answer: −53- \frac{5}{3}

m=y2−y1x2−x1=7−(2)−7−(−4)=−53m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{7 - (2)}{-7 - (-4)} = - \frac{5}{3}.

Common mistakes, and the answer each one gives:

  • Inverted the fraction (run over rise) → −35- \frac{3}{5}
  • Subtracted the coordinates in opposite orders → 53\frac{5}{3}
29

Answer: −12- \frac{1}{2}

Expand the bracket: 2x−10=6x−82 x - 10 = 6 x - 8. Collect the xx terms on one side and the numbers on the other, then divide: x=−12x = - \frac{1}{2}.

Common mistakes, and the answer each one gives:

  • Multiplied only the xx by the number outside the bracket → 34\frac{3}{4}
  • Moved a term across without changing its sign → −132- \frac{13}{2}
30

Answer: −15-15

Cross-multiply to clear both denominators: 3(x−9)=4(x−3)3\left(x - 9\right) = 4\left(x - 3\right). Expand, collect the xx terms and divide: x=−15x = -15.

Common mistakes, and the answer each one gives:

  • Multiplied only the numerators by the other denominator → −24-24
  • Added the fractions instead of cross-multiplying → 397\frac{39}{7}