Chapter 8 - Linear Equation & Coordinate geometry - Worked Examples

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Unit 8.1: Linear Equation In Two Variables

An equation is said to be linear equation in two variables if it is written in the form of:

ax + by + c = 0

where a, b & c are real numbers and the coefficients of x and y, i.e., a and b respectively, are not equal to zero.

Worked Example 1: Three times of one number is added to four times of other number The result obtained is 53. Represent this statement in the form of a linear equation.
Solution:
Let x be one number and y be another number.
Three times of x = 3x
Four times of y = 4y
Hence, 3x + 4y = 53 is a linear equation with two variables x and y.

MORE EXAMPLES:

Construct a linear equation with two variables for the given statements.
Example 1:
10x + 4y = 3 and
-x + 51y = 2
are linear equations in two variables.

Example 2:
Ahmed runs a stationary shop. He sold 25 notebooks and 30 registers for PKR 2400, in the month of January.
Solution:
We can express the above statement in a generalised form as given below.
Let x stand for one notebook. 25 notebooks mean 25x
Let y stand for one register. 30 registers mean 30y.
Hence, 25x + 30y = PKR 2400 is a linear equation with two variables x and y.


Example 3:
Six chairs and four tables cost PKR 25 000.
Solution:
The equation will be:
⇒ 6a + 4b = PKR 25 000,
where a stands for one chair,
and b stands for one table.

Example 4:
The difference of one-fourth of a number and one- fifth of another number is 1.
Solution:
The equation will be:

Example 5:
A number when added to 32 makes 100.
Solution:
⇒ x + 32 = 100 Ans.

Example 6:
A number multiplied by 4 and 3 taken away from the product gives the answer 9.
Solution:
⇒ 4x - 3 = 9 Ans.

Unit 8.3: Graph Of Linear Equation

Linear Equation in One Variable:

A linear equation in one variable is of the form:

• ax + b = 0.
• a + by = 0.

Here, a & b are numbers and x & y are variable. a & b are coefficient of x & y respectively.

Line Graph:
A linear equation can be represented as a line graph.
In order to draw the line graph we require several pairs of coordinates. These coordinates represent the relationship given in the equation.

Graph of ax + b = 0 Or ay + b = 0
The solution of ax + b = 0 is unchanged if any number is added, subracted, multiplied, or divided on both side of the equation. It means there is only one solution of the equation. On a graph, it appears to be a straight line either horizontal or vertical.

1) Graph of ax + b = 0
Worked Example 2:
(i) Let us draw a line of linear equation x + 6 = 10 on the graph.
Solution:
⇒ x + 6 = 10
Then x = 10 - 6 = 4
or x = 4
The graph of x = 4 is a vertical line.

2) Graph of a + by = 0
(ii) Let us draw a line of linear equation 7 + y = 9 on the graph.
Solution:
⇒ 7 + y = 9
Then y = 9 - 7 = 2
or y = 2
The graph of y = 2 is a horizontal line.

Linear Equation in OneTwo Variable:

If an ordered pair (x, y) is a solution to a linear equation in two variables, then it lies on the graph of the equation.
[Note: Ordered pair  is always written in small bracket & first value is always 'x' & second value is always 'y' i.e.. (x, y)]
A graph of a linear equation in two variable (ax + by = c) is a straight line.

Example 1:
For y = 3x, find its ordered pairs.
Solution:
⇒ y = 3x (The y value is always equal to '3 times of the x value).
Let x = 1
⇒ y = 3 x 1 = 3
∴ Ordered Pair = (1, 3)

Let x = 2
⇒ y = 3 x 2 = 6
∴ Ordered Pair = (2, 6)

Let x = 5
⇒ y = 3 x 5 = 15
∴ Ordered Pair = (5, 15)

Answer: (1, 3), (2, 6) and (5, 15) are all coordinates on the line y = 3x.

Example 2:
Find the coordinate or ordered pairs of the equation y = 3x + 1 and plot these coordinates on graph paper.
Solution:
We use substitution to calculate the values.
For y = 3x + 1
We replace the value of x for different numbers and record the y value.
Let x = 0
⇒ y = (3 x 0) + 1
⇒ y = 0 + 1 = 1
∴ Ordered Pair = (0, 1)

Let x = 1
⇒ y = (3 x 1) + 1
⇒ y = 3 + 1 = 4
∴ Ordered Pair = (1, 4)

Let x = 2
⇒ y = (3 x 2) + 1
⇒ y = 6 + 1 = 7
∴ Ordered Pair = (2, 7)

For convenience, a table of values is used to create the coordinates.

x	0	1	2
y	1	4	7

We have coordinates of the equation as (0, 1), (1, 4), (2, 7). Now we plot these coordinates on the graph paper.

Worked Example 3: Find the coordinates of point P located on the graph.
Solution:

• Begin at the point P and draw a dotted line either up or down the x-axis. The line to the point on x-axis is the x-coordinate.
• Similarly, begin at point P and draw a touching d form P dotted line touching to the point on y-axis is y-coordinate.
• In the given graph vertical line from P touches x-axis at x = 2
• Horizontal line from P touches y-axis at y = 4

∴ The coordinates of P(x, y) = (2, 4) Answer.