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Simultaneous equations graphically practice questions with answers

Foundation and Higher · Grade 5 · Non-calculator

A second set of simultaneous equations graphically questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

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Question 1 Easy

Do the lines meet: y=x+1 and y=5x?

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Answer: yes

  1. Gradients are 1 and 5.
  2. Different slopes meet.
  • Method (1): Gradients are 1 and 5.
  • Accuracy (1): Different slopes meet.

Question 2 Easy

Do the lines meet: y=2x+3 and y=2x+1?

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Answer: no

  1. Both have gradient 2.
  2. Same gradient means parallel.
  • Method (1): Both have gradient 2.
  • Accuracy (1): Same gradient means parallel.

Question 3 Medium

Find the coordinates of the point where the lines y=x+3 and y=4x meet.

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Answer: (1, 4)

  1. Equate: x + 3 = 4x, so 3x = 3 and x = 1.
  2. Calculate y: 4(1) = 4.
  • Method (1): Equate: x + 3 = 4x, so 3x = 3 and x = 1.
  • Accuracy (1): Calculate y: 4(1) = 4.

Question 4 Medium

Find the coordinates of the point where the lines y=3x+1 and y=x+5 meet.

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Answer: (2, 7)

  1. Equate: 3x + 1 = x + 5, so 2x = 4 and x = 2.
  2. Calculate y: 2 + 5 = 7.
  • Method (1): Equate: 3x + 1 = x + 5, so 2x = 4 and x = 2.
  • Accuracy (1): Calculate y: 2 + 5 = 7.

Question 5 Hard

How many solutions do y=2x+1 and y=2x+4 have?

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Answer: 0

  1. Equal gradient 2 with different intercepts.
  2. Parallel lines have 0 intersections.
  • Method (1): Equal gradient 2 with different intercepts.
  • Accuracy (1): Parallel lines have 0 intersections.

Question 6 Hard

How many solutions do y=3x+1 and y=x+2 have?

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Answer: 1

  1. Gradients 3 and 1 differ.
  2. Two lines with different gradients meet once.
  • Method (1): Gradients 3 and 1 differ.
  • Accuracy (1): Two lines with different gradients meet once.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Which line is steeper, y=6x-2 or y=2x+9?

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Answer: y=6x-2

  1. The gradient of y=6x-2 is 6 and the gradient of y=2x+9 is 2.
  2. Since 6 > 2, y=6x-2 is steeper.
  • Method (1): Gradients 6 and 2 identified
  • Accuracy (1): y=6x-2

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

What does it mean if two lines have equal gradients and different intercepts?

  1. They are parallel and never meet
  2. They meet at the origin
  3. They are perpendicular
Show answer

They are parallel and never meet Equal gradient means same slope; different intercepts mean separate lines.

Check 2

Do the lines y=2x+1 and y=2x+4 meet?

  1. no
  2. yes
  3. only at (0, 1)
Show answer

no Both have gradient 2, so they are parallel.

Check 3

Do the lines y=3x+1 and y=2x+4 meet?

  1. yes
  2. no
  3. at two points
Show answer

yes Gradients 3 and 2 are different, so they intersect.

Check 4

Where do y = x + 1 and y = 2x meet?

  1. (1, 2)
  2. (2, 1)
  3. (0, 1)
Show answer

(1, 2) x + 1 = 2x gives x = 1, then y = 2.

Check 5

Where do y = x + 2 and y = 3x meet?

  1. (1, 3)
  2. (2, 6)
  3. (3, 1)
Show answer

(1, 3) x + 2 = 3x gives 2x = 2, so x = 1, y = 3.

Check 6

Which property determines the steepness of a line?

  1. The gradient
  2. The y-intercept
  3. The length of the line
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The gradient Gradient measures vertical change per unit horizontal change.

Check 7

Which line is steeper: y = 3x + 1 or y = x + 4?

  1. y=3x+1
  2. y=x+4
  3. They are equally steep
Show answer

y=3x+1 Gradient 3 is greater than gradient 1.

Check 8

Which line is steeper: y = 5x + 1 or y = 2x + 4?

  1. y=5x+1
  2. y=2x+4
  3. They have the same slope
Show answer

y=5x+1 Gradient 5 is greater than gradient 2.

Check 9

How many intersections do distinct parallel lines y=2x+1 and y=2x+4 have?

  1. 0
  2. 1
  3. 2
Show answer

0 Parallel distinct lines have no points in common.

Check 10

How many solutions do y=3x+1 and y=x+2 have?

  1. 1
  2. 0
  3. 2
Show answer

1 Two distinct lines with different gradients intersect at exactly one point.

Check 11

Does a larger y-intercept make a line steeper?

  1. No, steepness depends only on gradient
  2. Yes, higher intercept means steeper
  3. Only if gradient is positive
Show answer

No, steepness depends only on gradient The intercept only determines where the line crosses the y-axis.

Check 12

Where do y = 2x + 1 and y = x + 4 meet?

  1. (3, 7)
  2. (1, 4)
  3. (2, 5)
Show answer

(3, 7) 2x + 1 = x + 4 gives x = 3, y = 7.

Check 13

Where do y = x + 3 and y = 4x meet?

  1. (1, 4)
  2. (3, 4)
  3. (2, 8)
Show answer

(1, 4) x + 3 = 4x gives 3x = 3, so x = 1, y = 4.

Check 14

Can two different straight lines intersect more than once?

  1. No, at most once
  2. Yes, up to twice
  3. Yes, infinitely many times
Show answer

No, at most once Two straight lines can intersect at 0 points (parallel) or 1 point.

Check 15

How do you algebraically find where two lines y = f(x) and y = g(x) meet?

  1. Equate f(x) = g(x) and solve for x
  2. Add f(x) and g(x)
  3. Multiply f(x) by g(x)
Show answer

Equate f(x) = g(x) and solve for x At the intersection, y-coordinates are equal.

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