krit.club logo

Coordinate Geometry - Equation of a Straight Line (y = mx + c)

Grade 9IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

The gradient (m) represents the steepness of the line, calculated as the 'rise over run'.

•

The y-intercept (c) is the point where the line crosses the y-axis (where x = 0).

•

Parallel lines have the same gradient (m1=m2m_1 = m_2).

•

A positive gradient slopes upwards from left to right; a negative gradient slopes downwards.

•

Horizontal lines have the equation y=ky = k (gradient is 0).

•

Vertical lines have the equation x=kx = k (gradient is undefined).

📐Formulae

y=mx+cy = mx + c (Slope-intercept form)

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} (Gradient formula)

y−y1=m(x−x1)y - y_1 = m(x - x_1) (Point-gradient form)

m1×m2=−1m_1 \times m_2 = -1 (Condition for perpendicular lines)

💡Examples

Problem 1:

Find the gradient and y-intercept of the line with equation 3x+2y=83x + 2y = 8.

Solution:

2y=−3x+8⇒y=−32x+42y = -3x + 8 \Rightarrow y = -\frac{3}{2}x + 4. Therefore, m=−1.5m = -1.5 and c=4c = 4.

Explanation:

To find mm and cc, rearrange the equation into the form y=mx+cy = mx + c by isolating yy on one side.

Problem 2:

Find the equation of the line passing through the points A(2,5)A(2, 5) and B(4,9)B(4, 9).

Solution:

m=9−54−2=42=2m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2. Substitute into y=mx+cy = mx + c: 5=2(2)+c⇒5=4+c⇒c=15 = 2(2) + c \Rightarrow 5 = 4 + c \Rightarrow c = 1. Equation: y=2x+1y = 2x + 1.

Explanation:

First, calculate the gradient using the two-point formula. Then, substitute one of the points and the gradient into the general equation to solve for cc.

Problem 3:

Find the equation of a line parallel to y=3x−5y = 3x - 5 that passes through the point (1,7)(1, 7).

Solution:

Parallel lines have the same gradient, so m=3m = 3. Using y−y1=m(x−x1)y - y_1 = m(x - x_1): y−7=3(x−1)⇒y−7=3x−3⇒y=3x+4y - 7 = 3(x - 1) \Rightarrow y - 7 = 3x - 3 \Rightarrow y = 3x + 4.

Explanation:

Identify that the gradient must be 3 because the lines are parallel. Then use the point-gradient formula or y=mx+cy=mx+c to find the new y-intercept.