Equation of a Straight Line

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Form 4 Chapter 5: The Straight Line


Step 1: Find the value of x, when y = 0.

When y = 0,

0 = x + 2 x = – 2

Example:

Draw the graph ofy = x + 2.

Given an equation of the form y = mx + c

Solution:

x – 2 – 1 1

y

1

2

OO


Example:

Step 2: Find the value of y, when x = 0.


Solution:

y

When x = 0,

y = 0 + 2

= 2

x – 1 1 – 2

y

1

2

OO




Example:

Step 3:Draw a straight line which passes through the two points.

Solution:

Remarks: You can choose any other two suitable points.

x – 1 1 – 2

y

1

2

OO



Determining whether a given point lies on a straight line

If a point lies on a straight line,

the coordinates of the point satisfy the

equation of the straight line.


Example

Solution:2y = 5x + 8

Substitute y = 9

2(9) = 18

Substitute x = 2

5(2) + 8 = 18

Determine whether the point (2, 9) lies on the straight line 2y = 5x + 8.

The same


Example

Conclusion:Left hand side = Right hand side

The point (2, 9) lies on the straight line 2y = 5x + 8.

x = 2 and y = 9 satisfy the equation 2y = 5x + 8.

Determine whether the point (2, 9) lies on the straight line 2y = 5x + 8.


Given the gradient and y-intercept

y = mx + c

Gradient

y-intercept


Example I:

y = x

Write the equation of the straight line given:

Gradient = 3, y-intercept = – 2

Gradientm = 3

y-interceptc = – 2

33 – 2 – 2


Example II:

y = x + 7

Gradient = and passes through the point (0, 7)4

1

4

1

Write the equation of the straight line given:

y-intercept(0, c) = (0, 7)

c = 7

Gradientm = 1

4

+ 7 + 7


Given the equation in the form y = mx + c

y = – 2x + 5

Determine the gradient and the y-intercept of the straight line:

Gradientm = – 2

y-intercept

c = constant term = 5


Given the equation of the form ax + by = c

3x + 5y = 15


5y = – 3x + 15

y = Make y as the subject

Write in the form:y = mx + c

– 3x + 15

5

3

5y = – x + 3


Given the equation of the form ax + by = c


Gradient

m = –

y-intercept

c = 3

3

5y = – x + 3

3

5


Straight line that is parallel to the x-axis

State the equation of the straight line AB.

y

2

4

x – 4 – 2 2

A B

Answer: y = 3

y-intercept = 3

If the y-intercept of a straight line which is parallel to the x-axis is k, then the equation of the straight line is y = k.

O


Straight line that is parallel to the y-axis

State the equation of the straight line AB.

y

2

4

x – 4 – 2 2

A

B

Answer: x = – 5

x-intercept = – 5

If the x-intercept of a straight line which is parallel to the y-axis is h, then the equation of the straight line is x = h.

O


A straight line that passes through a given point and has a specific gradient

Find the equation of the straight line which passes through the point (– 1, 2) and has a gradient of 2.

Solution:

Step 1: Substitute the value of m.

y = 2x + c


Solution:

Step 2: Substitute the x-coordinate and the y-coordinate into the equation to find c.

2 = 2(–1) + c c = 4




Solution:

Step 3: Write the equation with the values of m and c.

y = 2x + 4




Find the equation of the straight line which passes through point (1, 2) and point (3, 8).

Solution:

Step 1: Find the gradient, m, from the formula:

12

12

xx

yym

13

28

m m = 3

A straight line that passes through two given points



Solution:

Step 2: Substitute the value of m.

y = 3x + c




Solution:

Step 3: Substitute the x-coordinate and the y-coordinate from either point into the equation to find c.

2 = 3(1) + c c = –1




Solution:

Step 4: Write the equation with the values of m and c.

y = 3x – 1



The point of intersection of two straight lines

Find the point of intersection of two straight lines y = 3x + 1 and x + y = –3.

y

2

4

x – 4 – 2 2

x + y = –3

y = 3x + 1

From the graph, the point of intersection is (–1, –2).

O

Graphical Method


Step 1:

Find the point of intersection of two straight lines y = 3x + 1 and x + y = –3.

x + 3x + 1 = –34x = –4

x = –1

Substitute y = 3x + 1 into equation y = 3x + 1 … … x + y = –3 … …

Solving simultaneous equation



Find the point of intersection of two straight lines y = 3x + 1 and x + y = –3:

Substitute x = –1 into equation y = 3(–1) + 1 = –2



Step 2:


Find the point of intersection of two straight lines y = 3x + 1 and x + y = –3:

The point of intersection of the two straight lines

is (–1, –2).



Answer

Equation of a Straight Line

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