7760lecture 3 Notes

Chapter 2: Linear Equation with 18/02/2021
Two Variables
A linear equation involving two variables x and y has
the standard form
ax + by = c
where “a” and “b” are coefficient and “c” is constant
Chapter 2: Linear Equation with 18/02/2021
Two Variables
There are three main forms of linear equations.
Ax + By = C y = mx + b 2 2
where “a” and “b” where “m” is slope and where “m” is slope and
are coefficient and “b” is the y-intercept (x1 , y1) is a point on the
“c” is constant line
18/02/2021
Chapter 2: Linear Equation with Two Variables
Definition of x- and y-Intercepts

The x-intercept is the point where the
graph of a line intersects the x-axis.
The y-intercept is the point where the

graph of a line intersects the y-axis.
1. Standard Form
Example: Find the x- and y-intercepts: −3x+2y=12.
Therefore, the x-intercept is (−4, 0) Hence the y-intercept is (0, 6)
Answer: x-intercept: (−4, 0); y-intercept: (0, 6)

18/02/2021
Chapter 2: Linear Equation with Two Variables
Exercise 2.2 Page 51

Q.10. (x+y)/2 = 3x – 2y + 16
Q.12. –3x + 4y – 10 = 7x – 2y + 50
Q.14. (x – 2y)/3 – 12 = (2x + 4y)/3
Slope of straight-line graphs
The Slope of a line is a measure of how steep
the line is.
The slope of a line as the ratio of the vertical
change to the horizontal change. It is sometimes
referred to as the “rate of change” between two
points.
The letter “m” is used to represent slope.
The slope of a line can be positive,
negative, zero or undefined.
an upwards slope a horizontal line a downwards slope a vertical slope
y y y y
x x x x
Positive Slope Zero Slope Negative Slope Undefined slope
If a line is vertical its Slope cannot by specified.

Finding the slope from two given points
If we are given any two points (x1, y1) and (x2, y2) on a line we can
calculate the slope of the line as follows:
y
change in y (x2, y2)
the Slope, m =
change in x
y2 – y1
(x1, y1)
Draw a right-angled triangle
x2 – x1
between the two points on
the line as follows:
y2 – y1 x
the Slope, m =
x2 – x1
The Slope , m, is also known as Two-Point Formula

Example 2: Find the slope of the line passing through
(−3, −5) and (2, 1).
Solution: m
m
m
When using the slope formula, take care to be
consistent since order does matter.
You must subtract the coordinates of the first

point from the coordinates of the second
point for both the numerator and the
denominator in the same order.
2. The Slope – Intercept Form
The general equation of a straight line can be written as:
y = mx + b
The value of “m” tells us the Slope of the line.
The value of “b” tells us where the line crosses the y-axis.
This is called the y-intercept and it has the coordinate (0, b).
For example, the line y = 3x + 4

A slope of 3 and crosses the y-axis at the point (0, 4).
Example: Determine the slope and y-intercept:
y= +7.
Solution:
Answer:
The y-intercept is (0, 7),
and the slope is

It is not always the case that the
linear equation is given in slope-
intercept form.
When it is given in standard form,

you have to first solve for y to obtain
slope-intercept form.
Example: Express 3x+5y=30 in slope-intercept form and then
identify the slope and y-intercept.
Solution:
Answer: Slope-intercept form:
y-intercept: (0, 6); slope:

18/02/2021
Exercise 2.3 Page 54-55
Re-write each equation slope-intercept form and
determine the slope and y-intercept
Q.16. 3y – 5x + 20 = 4x – 2y +5
Q.17. 2x + 3y = 4x + 3y
Q.18. –5x + y – 12 = 2y – 5x
3. Point-Slope Form
Point-slope form emphasises the slope of the line and a point on
the line (that is not the y-intercept)
By rearranging Slope formula:
We get Point Slope Formula

Example: Find the y-intercept given the slope and a point
Example: Write an equation in slope intercept-form given two points.
Note: It doesn’t matter which

point you chose to construct
m the equation as long as the slope
is the same, and the point
selected must lie on the line.
m
18/02/2021
Exercise 2.4 Page 63
Determine the slope-intercept form of the linear equation
Q.4. Slope = –5/2, y-intercept = (0, – 20)
Q.8. Slope = 5, (-3, 12) lies on the line
Q.11. Slope = 2.5, (-2, 5) lies on the line
Q.23. (20, 240) and (15, 450) lie on line
Q.26. (5.76, -2.48) and (3.74, 8.76) lie on line
SYED JAWED IQBAL

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Chapter 2: Linear Equation with 18/02/2021

Definition of x- and y-Intercepts

The y-intercept is the point where the

Therefore, the x-intercept is (−4, 0) Hence the y-intercept is (0, 6)

Answer: x-intercept: (−4, 0); y-intercept: (0, 6)

Exercise 2.2 Page 51

Positive Slope Zero Slope Negative Slope Undefined slope

If a line is vertical its Slope cannot by specified.

The Slope , m, is also known as Two-Point Formula

You must subtract the coordinates of the first

For example, the line y = 3x + 4

and the slope is

When it is given in standard form,

Answer: Slope-intercept form:

y-intercept: (0, 6); slope:

We get Point Slope Formula

Note: It doesn’t matter which

SYED JAWED IQBAL

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