Parallel Lines (Gradients)

\( m_1=m_2 \)
Coordinate Geometry GCSE
Question 6 of 20
← All formulas ⟵ Prev Next ⟶

\( Is line through (0,0) and (2,2) parallel to y=x+1? \)

Hint (H)
Find gradients.

Explanation

Show / hide — toggle with X

Statement

Two lines in a plane are parallel if they never meet, no matter how far they are extended. In coordinate geometry, the condition for parallel lines is that their gradients (slopes) are equal. If line 1 has gradient \(m_1\) and line 2 has gradient \(m_2\), then:

\[ m_1 = m_2 \]

This means both lines rise and run at the same rate, so they never intersect (unless they are coincident — exactly the same line).

Why it’s true

  • The gradient measures the steepness of a line, defined as \(\Delta y / \Delta x\).
  • If two lines have the same gradient, their steepness is identical, so their directions are the same.
  • Different intercepts mean they are distinct parallel lines; the same intercept means they are the same line.

Recipe (how to use it)

  1. Find the gradient of each line (from equation or points).
  2. Compare gradients.
  3. If equal, the lines are parallel.
  4. If not equal, the lines intersect.

Spotting it

Parallel lines are often written in the form \(y = mx + c\). If the value of \(m\) is the same, the lines are parallel. Sometimes questions ask you to find a line parallel to another — in this case, keep the same gradient but change the intercept.

Common pairings

  • Equations of straight lines: “Find the equation of the line parallel to \(y=3x+1\) through a given point.”
  • Geometry problems involving parallel lines and transversals.
  • Coordinate proofs of parallelograms (opposite sides are parallel).

Mini examples

  1. Given: \(y=2x+3\) and \(y=2x-5\). Find: Are they parallel? Answer: Yes, both have gradient 2.
  2. Given: Line through (0,1) and (2,5). Gradient = \((5-1)/(2-0)=2\). Another line has equation \(y=2x+7\). Find: Are they parallel? Answer: Yes, both gradients equal 2.

Pitfalls

  • Forgetting to simplify gradients correctly when using two points.
  • Confusing parallel with perpendicular (where gradients multiply to \(-1\)).
  • Not distinguishing between parallel lines and coincident lines (same gradient and same intercept).

Exam strategy

  • Always calculate gradients carefully; use fractions if necessary.
  • When asked for a parallel line equation, copy the gradient and use the given point to find the intercept.
  • If proving a shape is a parallelogram, check both pairs of opposite sides for equal gradients.

Summary

Parallel lines have equal gradients. If two lines in the form \(y=mx+c\) share the same \(m\), they will never meet unless they are the same line. This concept is key in line equations, proofs, and coordinate geometry.

Worked examples

Show / hide (10) — toggle with E
  1. \( Are y=2x+3 and y=2x-5 parallel? \)
    1. \( Line 1 gradient = 2 \)
    2. \( Line 2 gradient = 2 \)
    3. \( m1=m2 \)
    4. Lines are parallel.
    Answer: Parallel
  2. \( Are y=3x+4 and y=-3x+1 parallel? \)
    1. \( Line 1 gradient = 3 \)
    2. \( Line 2 gradient = -3 \)
    3. Not equal → not parallel.
    Answer: Not parallel
  3. Find the gradient of line through (0,1) and (2,5).
    1. \( m=(5-1)/(2-0)=4/2=2 \)
    Answer: 2
  4. \( Is line through (1,2) and (3,6) parallel to y=2x-1? \)
    1. \( Gradient between (1,2) and (3,6) = (6-2)/(3-1)=4/2=2 \)
    2. Equation given has gradient 2
    3. \( m1=m2 → parallel. \)
    Answer: Parallel
  5. \( Find equation of line parallel to y=4x+1 through (0,3). \)
    1. \( Parallel line has same gradient m=4 \)
    2. \( Equation form y=4x+c \)
    3. \( Substitute (0,3): 3=4*0+c → c=3 \)
    4. \( Equation: y=4x+3 \)
    Answer: \( y=4x+3 \)
  6. \( Are y=1/2x+7 and line through (0,0),(2,1) parallel? \)
    1. \( Line 1 gradient=1/2 \)
    2. \( Line 2 gradient=(1-0)/(2-0)=1/2 \)
    3. Equal gradients → parallel.
    Answer: Parallel
  7. \( Equation of line parallel to y=-3x+2 through (2,1). \)
    1. \( Parallel gradient m=-3 \)
    2. \( Equation: y=-3x+c \)
    3. \( Substitute (2,1): 1=-6+c → c=7 \)
    4. \( Equation: y=-3x+7 \)
    Answer: \( y=-3x+7 \)
  8. Find gradient of line parallel to vector (2,5).
    1. \( Gradient = rise/run = 5/2 \)
    Answer: 5/2
  9. \( Show that lines y=2x+1 and y-4=2(x-3) are parallel. \)
    1. \( First line gradient=2 \)
    2. \( Second line: y-4=2x-6 → y=2x-2 \)
    3. \( Gradient=2 \)
    4. Both equal → parallel.
    Answer: Parallel
  10. \( Find k such that line through (0,2) and (k,6) is parallel to y=2x+1. \)
    1. \( Gradient = (6-2)/(k-0)=4/k \)
    2. \( Parallel means 4/k=2 \)
    3. \( Solve: k=2 \)
    Answer: \( k=2 \)