CIE IGCSE Mathematics (0580) Equations of linear graphs Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Types of number Study Notes
LEARNING OBJECTIVE
- Interpreting and Finding the Equation of a Straight Line
Key Concepts:
- Equation of a Straight Line
Interpreting and Finding the Equation of a Straight Line
Interpreting and Finding the Equation of a Straight Line
A straight-line graph is most commonly written in the form:
\( y = mx + c \)
Where:
- \( m \) is the gradient (slope) of the line
- \( c \) is the y-intercept (the value of y when \( x = 0 \))
Key Forms:
- y = mx + c :
- a standard linear equation
- x = c :
- a vertical line crossing the x-axis at \( x = c \) (gradient is undefined)
- y = k :
- a horizontal line crossing the y-axis at \( y = k \) (gradient = 0)
- ax + by = c :
- general linear form that can be rearranged into \( y = mx + c \); useful for algebraic manipulation or when working with systems of equations.
Steps to Find the Equation of a Line (From a Graph or Two Points):
- Find the gradient \( m \) using two points: \( m = \frac{y_2 – y_1}{x_2 – x_1} \)
- Substitute \( m \) and one point into \( y = mx + c \) to solve for \( c \)
- Write the final equation in the form \( y = mx + c \)
Example :
Find the gradient and y-intercept of the equation \( y = 6x + 3 \).
▶️ Answer/Explanation
This is already in the form \( y = mx + c \)
So, Gradient \( m = 6 \), y-intercept \( c = 3 \)
Answer: Gradient = 6, y-intercept = 3
Example :
A line passes through the points (1, 2) and (3, 6). Find its equation.
▶️ Answer/Explanation
Gradient \( m = \frac{6 – 2}{3 – 1} = \frac{4}{2} = 2 \)
Use point (1, 2) in \( y = mx + c \): \( 2 = 2(1) + c \Rightarrow c = 0 \)
Answer: \( y = 2x \)
Example :
A graph is a vertical line crossing the x-axis at \( x = -3 \). What is its equation?
▶️ Answer/Explanation
Answer: \( x = -3 \)
Example :
Find the equation of the line passing through the points \( (1, 2) \) and \( (3, 6) \) in the form \( ax + by = c \).
▶️ Answer/Explanation
Step 1: Find gradient
\( m = \frac{6 – 2}{3 – 1} = \frac{4}{2} = 2 \)
Step 2: Use point-slope form: \( y – y_1 = m(x – x_1) \)
\( y – 2 = 2(x – 1) \Rightarrow y = 2x – 2 + 2 = 2x \)
This gives us: \( y = 2x \)
To write in general form: move terms to one side → \( 2x – y = 0 \)
Answer: \( 2x – y = 0 \)
Example :
Find the equation of the line given the graph:
▶️Answer/Explanation
Identify the y-intercept \( b \)
The graph passes through the point \( (0, 4) \), so the y-intercept is:
\(c = 4 \)
Use two points on the line: \( (0, 4) \) and \( (4, 2) \).
The slope is calculated as:
\( m = \frac{\text{rise}}{\text{run}} = \frac{2 – 4}{4 – 0} = \frac{-2}{4} = -\frac{1}{2} \)
The slope-intercept form is: \( y = mx + c \)
Substitute \( m = -\frac{1}{2} \) and \( c = 4 \):
\( y = -\frac{1}{2}x + 4 \)