Intersection of two lines

Find the coordinates of the point where two given lines intersect using algebraic methods.

42 questions · Moderate -0.7

1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships
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Edexcel C1 Q10
9 marks Moderate -0.8
The straight line \(l_1\) has equation \(4y + x = 0\). The straight line \(l_2\) has equation \(y = 2x - 3\).
  1. On the same axes, sketch the graphs of \(l_1\) and \(l_2\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. [3]
The lines \(l_1\) and \(l_2\) intersect at the point \(A\).
  1. Calculate, as exact fractions, the coordinates of \(A\). [3]
  2. Find an equation of the line through \(A\) which is perpendicular to \(l_1\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]
Edexcel C1 Q23
11 marks Moderate -0.8
The straight line \(l_1\) with equation \(y = \frac{3}{2}x - 2\) crosses the \(y\)-axis at the point \(P\). The point \(Q\) has coordinates \((5, -3)\). The straight line \(l_2\) is perpendicular to \(l_1\) and passes through \(Q\).
  1. Calculate the coordinates of the mid-point of \(PQ\). [3]
  2. Find an equation for \(l_2\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integer constants. [4]
The lines \(l_1\) and \(l_2\) intersect at the point \(R\).
  1. Calculate the exact coordinates of \(R\). [4]
Edexcel C1 Q27
10 marks Moderate -0.8
The points \(A\) and \(B\) have coordinates \((4, 6)\) and \((12, 2)\) respectively. The straight line \(l_1\) passes through \(A\) and \(B\).
  1. Find an equation for \(l_1\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(l_2\) passes through the origin and has gradient \(-4\).
  1. Write down an equation for \(l_2\). [1]
The lines \(l_1\) and \(l_2\) intercept at the point \(C\).
  1. Find the exact coordinates of the mid-point of \(AC\). [5]
Edexcel C1 Q5
9 marks Moderate -0.8
The straight line \(l_1\) has equation \(4y + x = 0\). The straight line \(l_2\) has equation \(y = 2x - 3\).
  1. On the same axes, sketch the graphs of \(l_1\) and \(l_2\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. [3]
The lines \(l_1\) and \(l_2\) intersect at the point \(A\).
  1. Calculate, as exact fractions, the coordinates of \(A\). [3]
  2. Find an equation of the line through \(A\) which is perpendicular to \(l_1\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]
Edexcel C1 Q8
10 marks Moderate -0.8
The points \(A\) and \(B\) have coordinates \((4, 6)\) and \((12, 2)\) respectively. The straight line \(l_1\) passes through \(A\) and \(B\).
  1. Find an equation for \(l_1\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(l_2\) passes through the origin and has gradient \(-4\).
  1. Write down an equation for \(l_2\). [1]
The lines \(l_1\) and \(l_2\) intercept at the point \(C\).
  1. Find the exact coordinates of the mid-point of \(AC\). [5]
Edexcel C1 Q8
10 marks Moderate -0.3
The points \(P\) and \(Q\) have coordinates \((7, 4)\) and \((9, 7)\) respectively.
  1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(m\) has gradient \(8\) and passes through the origin, \(O\).
  1. Write down an equation for \(m\). [1]
The lines \(l\) and \(m\) intersect at the point \(R\).
  1. Show that \(OP = OR\). [5]
Edexcel C1 Q8
11 marks Moderate -0.8
The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A(5, 3)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).
  1. Find the coordinates of \(B\). [3]
  2. Find the coordinates of the mid-point of \(AB\). [2]
  3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]
Edexcel C1 Q3
6 marks Moderate -0.8
The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\).
  1. Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [3]
  2. Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
Edexcel C1 Q10
14 marks Standard +0.3
The straight line \(l\) has gradient 3 and passes through the point \(A(-6, 4)\).
  1. Find an equation for \(l\) in the form \(y = mx + c\). [2]
The straight line \(m\) has the equation \(x - 7y + 14 = 0\). Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
  1. find the coordinates of \(B\) and \(C\), [4]
  2. show that \(\angle BAC = 90°\), [4]
  3. find the area of triangle \(ABC\). [4]
OCR C1 Q4
8 marks Moderate -0.8
The straight line \(l_1\) has gradient 2 and passes through the point with coordinates \((4, -5)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) is perpendicular to the line with equation \(3x - y = 4\) and passes through the point with coordinates \((3, 0)\).
  1. Find an equation for \(l_2\). [3]
  2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
OCR C1 Q7
11 marks Moderate -0.8
The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A (5, 3)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).
  1. Find the coordinates of \(B\). [3]
  2. Find the coordinates of the mid-point of \(AB\). [2]
  3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]
OCR C1 Q5
7 marks Moderate -0.8
The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\). \begin{enumerate}[label=(\roman*)] \item Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [4] \item Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
OCR MEI C1 Q5
4 marks Moderate -0.8
Find the coordinates of the point of intersection of the lines \(y = 3x - 2\) and \(x + 3y = 1\). [4]
OCR MEI C1 Q2
4 marks Moderate -0.8
Find the coordinates of the point of intersection of the lines \(y = 5x - 2\) and \(x + 3y = 8\). [4]
WJEC Unit 1 2019 June Q04
15 marks Easy -1.3
The line \(L_1\) passes through the points \(A(-1, 3)\) and \(B(2, 9)\). The line \(L_2\) has equation \(2y + x = 25\) and intersects \(L_1\) at the point \(C\). \(L_2\) also intersects the \(x\)-axis at the point \(D\).
  1. Show that the equation of the line \(L_1\) is \(y = 2x + 5\). [3]
    1. Find the coordinates of the point \(D\).
    2. Show that \(L_1\) and \(L_2\) are perpendicular.
    3. Determine the coordinates of \(C\). [5]
  3. Find the length of \(CD\). [2]
  4. Calculate the angle \(ADB\). Give your answer in degrees, correct to one decimal place. [5]
WJEC Unit 1 2022 June Q3
11 marks Moderate -0.8
The line \(L_1\) passes through the points \(A(0, 5)\) and \(B(3, -1)\).
  1. Find the equation of the line \(L_1\). [3]
The line \(L_2\) is perpendicular to \(L_1\) and passes through the origin \(O\).
  1. Write down the equation of \(L_2\). [1]
The lines \(L_1\) and \(L_2\) intersect at the point \(C\).
  1. Calculate the area of triangle \(OAC\). [4]
  2. Find the equation of the line \(L_3\) which is parallel to \(L_1\) and passes through the point \(D(4, 2)\). [2]
  3. The line \(L_3\) intersects the \(y\)-axis at the point \(E\). Find the area of triangle \(ODE\). [1]
WJEC Unit 1 2024 June Q7
11 marks Easy -1.2
  1. The line \(L_1\) passes through the points \(A(-3, 0)\) and \(B(1, 4)\). Determine the equation of \(L_1\). [3]
  2. The line \(L_2\) has equation \(y = 3x - 3\).
    1. Given that \(L_1\) and \(L_2\) intersect at the point C, find the coordinates of C.
    2. The line \(L_2\) crosses the \(x\)-axis at the point D. Show that the coordinates of D are \((1, 0)\). [4]
  3. Calculate the area of triangle \(ACD\). [2]
  4. Determine the angle \(ACD\). [2]