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GCSE · Algebra: straight-line graphs

5.15 Equation of a Line

EDEXCEL 1MA1 · Calculator allowed · about 72 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

The equation of a straight line is y = 3x + 7.

(a)Write down the gradient of the line.(1)

(b)Write down the coordinates of the point where the line crosses the y-axis.(1)

(Total for Question 1 is 2 marks)

A straight line has equation y = 5 - 2x.

(a)State the gradient of the line.(1)

(b)State the y-intercept of the line.(1)

(Total for Question 2 is 2 marks)

Point C has coordinates (2, 3) and point D has coordinates (8, 21). Find the gradient of the line CD.

(Total for Question 3 is 2 marks)

Which of these lines is parallel to the line with equation y = 4x - 1?

A) y = 4x + 5
B) y = -4x + 5
C) y = (1/4)x - 1
D) 4y = x - 8

(Total for Question 4 is 1 mark)

Line L has equation 2y = 6x + 8. Write the equation of L in the form y = mx + c, and state the gradient and the y-intercept of L.

(Total for Question 5 is 3 marks)

Find the equation of the straight line with gradient 3 that passes through the point (2, 5). Give your answer in the form y = mx + c.

(Total for Question 6 is 3 marks)

A straight line passes through the points (1, 4) and (3, 10). Find the equation of the line, giving your answer in the form y = mx + c.

(Total for Question 7 is 4 marks)

Answer the following about horizontal and vertical lines.

(a)Write down the equation of the straight line that is parallel to the x-axis and passes through the point (0, -3).(1)

(b)Write down the equation of the straight line that is parallel to the y-axis and passes through the point (5, 2).(1)

(Total for Question 8 is 2 marks)

P has coordinates (-2, 3) and Q has coordinates (6, -9). Find the coordinates of the midpoint of PQ.

(Total for Question 9 is 2 marks)

Line L1 has equation y = 2x + 3. Line L2 has equation 4x - 2y = 6. Determine whether L1 and L2 are the same line, are parallel but different lines, or are neither. Justify your answer.

(Total for Question 10 is 2 marks)

Find the equation of the straight line that is parallel to y = 4x - 3 and passes through the point (2, 1). Give your answer in the form y = mx + c.

(Total for Question 11 is 3 marks)

Line L has equation 3x + 2y = 12.

(a)Find the coordinates of the point where L crosses the x-axis.(2)

(b)Find the coordinates of the point where L crosses the y-axis.(2)

(Total for Question 12 is 4 marks)

A taxi firm uses the formula C = 2.50 + 1.20m to work out the cost, C pounds, of a journey of m miles. This formula is the equation of a straight line.

(a)State the gradient of the line and explain what it represents in this context.(2)

(b)State the y-intercept of the line and explain what it represents in this context.(2)

(Total for Question 13 is 4 marks)

Find the equation of the straight line that is perpendicular to y = (1/2)x + 4 and passes through the point (3, -1). Give your answer in the form y = mx + c.

(Total for Question 14 is 4 marks)

A has coordinates (-2, 1) and B has coordinates (4, 5). Find the length of AB, giving your answer correct to 3 significant figures.

(Total for Question 15 is 3 marks)

Line L1 has equation 4x + 2y = 10. Line L2 passes through the points (0, 3) and (2, 7). Determine, showing your working, whether L1 and L2 are perpendicular.

(Total for Question 16 is 4 marks)

Show that the point (4, 11) lies on the line with equation y = 3x - 1.

(Total for Question 17 is 2 marks)

Line L has equation 2x + 3y = 12. L crosses the positive x-axis at point A and the positive y-axis at point B. A triangle is formed by L and the positive x- and y-axes.

Diagram NOT accurately drawn

(a)Find the coordinates of A and the coordinates of B.(2)

(b)Find the area of the triangle formed by L and the positive x- and y-axes.(2)

(Total for Question 18 is 4 marks)

Line L is perpendicular to the line with equation y = -(1/3)x + 2 and passes through the point A(-1, 5).

(a)Find the gradient of L.(2)

(b)Find the equation of L, giving your answer in the form y = mx + c.(2)

(c)L crosses the x-axis at point B. Find the coordinates of B.(2)

(d)Find the length of AB, giving your answer correct to 3 significant figures.(3)

(Total for Question 19 is 9 marks)

Mark scheme · 5.15 Equation of a Line

Question 1

(a) B1 3 cao
(a) Answer: 3
(b) B1 (0, 7) oe
(b) Answer: (0, 7)

Question 2

(a) B1 -2 cao
(a) Answer: -2
(b) B1 5 oe, e.g. (0, 5)
(b) Answer: 5

Question 3

M1 (21 - 3) / (8 - 2) oe correct substitution into the gradient formula
A1 3 cao
Answer: 3

Question 4

B1 A cao
Answer: A) y = 4x + 5

Question 5

M1 divides both sides by 2, e.g. y = 3x + 4 oe
A1 gradient = 3
A1 y-intercept = 4 oe (0, 4)
Answer: y = 3x + 4; gradient = 3; y-intercept = (0, 4)

Question 6

M1 substitutes m = 3 and (2, 5) into y = mx + c, e.g. 5 = 3(2) + c, or uses y - 5 = 3(x - 2)
M1 correctly solves for c, e.g. c = -1 (dep)
A1 y = 3x - 1 oe cao
Answer: y = 3x - 1

Question 7

M1 (10 - 4) / (3 - 1) oe correct method for gradient
A1 gradient = 3 cao
M1 substitutes a point and the gradient into y = mx + c to find c (dep)
A1 y = 3x + 1 oe cao
Answer: y = 3x + 1

Question 8

(a) B1 y = -3 cao
(a) Answer: y = -3
(b) B1 x = 5 cao
(b) Answer: x = 5

Question 9

M1 correct use of the midpoint formula ((x1+x2)/2, (y1+y2)/2)
A1 (2, -3) cao
Answer: (2, -3)

Question 10

M1 rearranges L2 to y = 2x - 3 oe
A1 cao: states L1 and L2 are parallel but different lines, since both have gradient 2 but different y-intercepts (3 and -3)
Answer: Parallel but different lines (both have gradient 2, but y-intercepts are 3 and -3)

Question 11

B1 gradient = 4, since parallel lines have equal gradients
M1 substitutes (2, 1) and m = 4 into y = mx + c (dep)
A1 y = 4x - 7 oe cao
Answer: y = 4x - 7

Question 12

(a) M1 sets y = 0 in 3x + 2y = 12
(a) A1 (4, 0) cao
(a) Answer: (4, 0)
(b) M1 sets x = 0 in 3x + 2y = 12
(b) A1 (0, 6) cao
(b) Answer: (0, 6)

Question 13

(a) B1 gradient = 1.20
(a) B1 oe, e.g. the cost increases by 1.20 pounds for each extra mile / the charge is 1.20 pounds per mile
(a) Answer: Gradient = 1.20; it represents the cost, in pounds, charged per mile
(b) B1 y-intercept = 2.50
(b) B1 oe, e.g. the fixed charge before any miles are travelled / the cost when m = 0
(b) Answer: y-intercept = 2.50; it represents the fixed charge, in pounds, before any miles are travelled

Question 14

M1 uses m1 x m2 = -1 to find the perpendicular gradient
A1 gradient = -2 cao
M1 substitutes (3, -1) and m = -2 into y = mx + c to find c (dep)
A1 y = -2x + 5 oe cao
Answer: y = -2x + 5

Question 15

M1 correct substitution into √(x2-x1)² + (y2-y1)²
A1 √52 oe (e.g. √36 + 16) seen
A1 awrt 7.21
Answer: 7.21

Question 16

M1 rearranges L1 to y = -2x + 5 oe to find gradient -2
M1 finds gradient of L2 = (7 - 3)/(2 - 0) = 2
A1 product of gradients = -2 x 2 = -4
A1 cao: concludes L1 and L2 are not perpendicular, since the product is not -1
Answer: Not perpendicular (product of gradients = -4)

Question 17

M1 substitutes x = 4 into y = 3x - 1
A1 cso: obtains y = 11, matching the given point, so (4, 11) lies on the line
Answer: y = 3(4) - 1 = 11, so (4, 11) lies on the line

Question 18

(a) M1 sets y = 0 to find A and x = 0 to find B
(a) A1 A(6, 0) and B(0, 4), both correct
(a) Answer: A(6, 0), B(0, 4)
(b) M1 uses area = (1/2) x base x height with base = 6 and height = 4, ft from part a
(b) A1 12 (square units) cao
(b) Answer: 12 square units

Question 19

(a) M1 uses m1 x m2 = -1 with m1 = -1/3
(a) A1 3 cao
(a) Answer: 3
(b) M1 substitutes A(-1, 5) and m = 3 (ft from part a) into y = mx + c
(b) A1 y = 3x + 8 oe cao
(b) Answer: y = 3x + 8
(c) M1 sets y = 0 in their equation from part b (ft)
(c) A1 (-8/3, 0) oe awrt (-2.67, 0), cao
(c) Answer: (-8/3, 0)
(d) M1 correct substitution of A(-1, 5) and their B (ft) into √(x2-x1)² + (y2-y1)²
(d) A1 √250/3 oe (e.g. √25/9 + 25) seen
(d) A1 awrt 5.27
(d) Answer: 5.27

Equation of a Line - Worksheets, Questions and Revision