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Equations of Straight Lines (y = mx + c)
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Foundational Skills
Identify the y-intercept from y = mx + c
\[ y = 2x + 5 \rightarrow c = 5 \]
Identify the value of c from an equation.
Identify gradient and y-intercept
\[ y = 3x – 4 \rightarrow m = 3, c = -4 \]
Identify both m and c from standard form.
Identify y-intercept when c = 0
\[ y = 4x \rightarrow c = 0 \]
Recognise y = mx has y-intercept of 0.
Identify from rearranged form
\[ y = 5 + 2x \rightarrow m = 2 \]
Identify m and c when written as y = c + mx.
Match equation to description
\[ m = 3, c = 2 \rightarrow y = 3x + 2 \]
Match a verbal description to an equation.
Understand meaning of y-intercept
\[ y = 2x + 5 \text{ crosses at } (0, 5) \]
Find coordinates where line crosses y-axis.
Find y-intercept by substitution
\[ x = 0: y = 3(0) + 7 = 7 \]
Verify y-intercept by substituting x = 0.
Writing Equations
Write equation (positive m and c)
\[ m = 2, c = 3 \rightarrow y = 2x + 3 \]
Write equation given positive m and c.
Write equation (negative gradient)
\[ m = -3, c = 4 \rightarrow y = -3x + 4 \]
Write equation when gradient is negative.
Write equation (negative y-intercept)
\[ m = 2, c = -5 \rightarrow y = 2x – 5 \]
Write equation when y-intercept is negative.
Write equation (both negative)
\[ m = -2, c = -3 \rightarrow y = -2x – 3 \]
Write equation when both m and c negative.
Write equation (fractional gradient)
\[ m = \frac{1}{2}, c = 4 \rightarrow y = \frac{1}{2}x + 4 \]
Write equation with a fractional gradient.
Write equation (through origin)
\[ m = 5, \text{origin} \rightarrow y = 5x \]
Write equation where c = 0.
Write equation (y-int as point)
\[ m = 3, (0, 4) \rightarrow y = 3x + 4 \]
Write equation when y-intercept given as (0, c).
Finding Equations from Graphs
Find equation from graph (positive m, c)
\[ \text{Read } m \text{ and } c \text{ from graph} \]
Write equation from graph (both positive).
Find equation from graph (negative m)
\[ \text{Line slopes down: } m < 0 \]
Write equation from graph with negative gradient.
Find equation from graph (negative c)
\[ \text{Y-intercept below } x\text{-axis} \]
Write equation from graph with negative c.
Find equation from graph (through origin)
\[ \text{Line through } (0, 0) \]
Write equation from graph where c = 0.
Find equation from graph (fractional m)
\[ \text{Gradient } = \frac{\text{rise}}{\text{run}} \]
Write equation from graph with fractional m.
Find equation from marked points
\[ (0, 2), (3, 8) \text{ marked} \]
Find equation using two marked points.
Finding Equations from Points
Find equation from gradient and point
\[ m = 2, (3, 7) \rightarrow y = 2x + 1 \]
Find c by substituting a point.
Find equation (negative m) from point
\[ m = -3, (2, 1) \rightarrow y = -3x + 7 \]
Find equation with negative gradient from point.
Find equation (fractional m) from point
\[ m = \frac{1}{2}, (4, 5) \rightarrow y = \frac{1}{2}x + 3 \]
Find equation with fractional gradient from point.
Find equation from two points (positive m)
\[ (1, 3), (3, 7) \rightarrow y = 2x + 1 \]
Find equation by calculating m first.
Find equation from two points (negative m)
\[ (1, 5), (3, 1) \rightarrow y = -2x + 7 \]
Find equation when gradient is negative.
Find equation from two points (negative coords)
\[ (-2, -1), (2, 7) \rightarrow y = 2x + 3 \]
Find equation with negative coordinates.
Find equation from two points (fractional m)
\[ (0, 1), (4, 3) \rightarrow y = \frac{1}{2}x + 1 \]
Find equation with fractional gradient.
Special Cases and Applications
Equation of horizontal line
\[ \text{Through } (3, 5) \rightarrow y = 5 \]
Recognise horizontal lines have equation y = c.
Equation of vertical line
\[ \text{Through } (4, 2) \rightarrow x = 4 \]
Recognise vertical lines have equation x = k.
Verify if point lies on line
\[ (2, 9) \text{ on } y = 3x + 3 \text{?} \]
Check if a point satisfies the equation.
Find where line crosses x-axis
\[ y = 2x + 6 \rightarrow (-3, 0) \]
Find x-intercept by setting y = 0.
Find y-value for given x
\[ y = 2x + 1, x = 3 \rightarrow y = 7 \]
Substitute x into equation to find y.
Find x-value for given y
\[ y = 2x + 1, y = 7 \rightarrow x = 3 \]
Rearrange to find x when y is given.
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