2.13.1 Linear Graph
Linear means straight and a graph is a diagram which shows a connection or relation between two
or more quantities. So, the linear graph is just a straight-line graph.
Equation of a straight line is
\( y=mx+c \)
Where \(m\) is the gradient and \( c \) is the \( y \)-intercept of the graph
To find equation of a straight line
Work out the gradient \( m \) if not given in the question
Take any point from the line or use any one from the question if given
Put these values in \( y=mx+c \) to get value of \( c \)
Now use value of \( m \) and \( c \) to represent the equation of the line
Worked example:
A line joins \( A(1, 3) \) to \( B(5, 8) \).
Find the equation of the line \( AB \).
Give your answer in the form \( y=mx+c \).
Work out the gradient by using \( m=\frac{y_2-y_1}{x_2-x_1} \)
\( m=\frac{8-3}{5-1} \)
\( m=\frac{5}{4} \)
Substitute the gradient and coordinates of one of the given two points and solve for \( c \)
\( 3=\frac{5}{4}\times1+c \)
\( 3-\frac{5}{4}=c \)
\( c=\frac{7}{4} \)
Substitute the value of gradient and \( y \)-intercept into the equation
\( y=\frac{5}{4}x+\frac{7}{4} \)
\( y=\frac{5}{4}x+\frac{7}{4} \)
To draw a linear graph
Rearrange the equation in the form \( y=mx+c \) if its not
Plot the point \( c \) on the \( y \)-axis
Go \( 1 \) across \( m \) up and plot the point
Join these two points by a straight line
Worked example:
On the axes below, draw the graph of \( 3x+5y=15 \).
Rearrange the equation in the form \( y=mx+c \)
\( 5y=-3x+15 \)
\( y=-\frac{3}{5}x+\frac{15}{5} \)
\( y=-\frac{3}{5}x+3 \)
Gradient is \( -\frac{3}{5} \) and \( y \)-intercept is \( 3 \)
Plot \( 3 \) on the \( y \)-axis and for every \( 5 \) across move \( 3 \) units down and plot a point
2.13.1.1 Parallel & Perpendicular lines
Parallel lines:
Two lines are said to be parallel if they have the same gradient.
Parallel lines never intersect
Parallel lines are always same distance apart
As parallel lines have the same gradient, a line in the form \( y=mx+c \) will be parallel to
\( y=mx+d \)
where \( m \) is the same for both lines and \( cd \)
To find the equation of a line parallel to another line
Use gradient of the given line and put the coordinates of the given point to find the
\( y \)-intercept
Worked example:
Find the equation of the line that is parallel to
\( y=5x+11 \) and passes through \( 0, 3 \).
As parallel lines have same gradient, so this line will be in the form
\( y=5x+d \)
Substitute \( (0, 3) \) in the equation and solve for \( d \)
\( 3=5(0)+d \)
\( d=3 \)
Put \( d=3 \) in the equation
\( y=5x+3 \)
\( y=5x+3 \)
Perpendicular lines:
Two lines are said to be perpendicular if they intersect each other at the right angle.
Gradients \( m_1 \) and \( m_2 \) are perpendicular if \( m_1 m_2 =-1 \)
To find perpendicular gradient use \( m_1 =-\frac{1}{m_2} \) called negative reciprocal.
Worked example:
Find the equation of the straight line that is perpendicular to the line \( y=\frac{1}{2}x+1 \) and passes through the point \( 1, 3 \).
Gradient of the line perpendicular to given equation is
\( m=-\frac{1}{\frac{1}{2}}=-2 \)
Substitute \( m=-2 \), and the point \( (1, 3) \) in the equation and solve for \( d \)
\( 3=-2\times1+d \)
\( 3+2=d \)
\( d=5 \)
Put \( m=-2 \) and \( d=5 \) in the equation
\( y=-2x+5 \)
\( y=-2x+5 \)
Test yourself
Question 1:
The coordinates of \( P \) are \( (-3, 8) \) and the coordinates of \( Q \) are \( (9, -2) \).
Find the equation of the perpendicular bisector of \( PQ \).
[4]
Find the equation of the line parallel to \( PQ \) that passes through the point \( (6, -1) \) .
[3]
Question 2:
A straight line joins the points \( A(-2, -3) \) and \( C(1, 9) \).
Find the equation of the line \( AC \) in the form \( y=mx+c \).
[3]
\( ABCD \) is a kite, where \( AC \) is the longer diagonal of the kite. \( B \) is the point \( ( 3.5, 2) \).
Find the equation of the line \( BD \) in the form \( y=mx+c \).
[3]
The diagonal \( AC \) and \( BD \) intersect at \( (-0.5, 3) \).
Work out the coordinates of \( D\).
[3]
Question 3:
The equation of line \( L \) is \( 3x-8y+20=0 \).
Find the gradient of line \( L \).
[2]
Find the coordinates of the point where line \( L \) cuts the \( y \) axis.
[1]
Question 4:
Write down the equation of the line parallel to line \( L \) that passes through the point \( B \).
[2]
Question 5:
The equation of a straight line is \( 2y=3x+4 \).
Find the gradient of this line.
[1]
Find the coordinates of the point where the line crosses the y axis.
[1]