Linear Equations in Two Variables.pptx

1. Linear Equations in
Two Variables

2. At the end of this session, you will be
able to
• distinguish standard form of linear equations from slope-
intercept form;
• write the linear equation 𝐴𝑥 + 𝐵𝑦 = 𝐶 in the form 𝑦 = 𝑚𝑥 + 𝑏
and vice versa;
• graphs a linear equation given (a) any two points; (b) the x-
and y-intercepts; (c) the slope and a point on the line; and
• describes the graph of a linear equation in terms of its
intercepts and slope.

3. LINEAR EQUATION IN TWO
VARIABLES
If 𝐴, 𝐵, and 𝐶 are real numbers, and if 𝐴 and 𝐵
are not both equal to 0, then 𝑨𝒙 + 𝑩𝒚 = 𝑪 is called
a linear equation in two variables.
The numbers 𝐴 and 𝐵 are the coefficients of
the variables 𝑥 and 𝑦, respectively, while the
number 𝐶 is the constant.
Page 134 on your Mathematics Book

4. The equation 𝑥 + 𝑦 = 19 is written in standard form
where 𝐴 = 1, 𝐵 = 1, and 𝐶 = 19.
So, when can we say that a linear equation is in its
standard form?
The standard form of a linear equation in two variables
is written in the order
𝑨𝒙 + 𝑩𝒚 =𝑪.

5. LINEAR EQUATION IN TWO
VARIABLES
A linear equation is an equation in two variables which can be
written in two forms.
1. STANDARD FORM: Ax + By = C, where A, B and C are real
numbers , A and B not both 0
Example: 3x + 5y = 21
2. SLOPE-INTERCEPT FORM: y = mx + b, where m is the slope
and b is the y-intercept, m and b are real numbers.
Example: y = 2x – 5 where m = 2 and b = -5
y =
𝟑
𝟓
𝐱 + 𝟐𝟏 where m =
𝟑
𝟓
and b = 21

6. Consider the equation 4𝑦 = 6 − 5𝑥.
Is the equation a linear equation in two variables?
YES
The equation 4𝑦 = 6 − 5𝑥 is a linear equation in two variables because:
1. it has two variables, 𝑥 and 𝑦;
2. it has only 1 variable in each term;
3. the exponent of the variable in each term is 1 which means the
degree of the equation is 1;
4. there is no variable in the denominator; and
5. there is no variable inside a radical sign.

7. Writing the Linear Equation
𝑨𝒙 + 𝑩𝒚 = 𝑪 in the Form 𝒚 = 𝒎𝒙 + 𝒃
and Vice versa
How do we rewrite the equation 𝑨𝒙 + 𝑩𝒚 = 𝑪
in the form y = mx + b?

8. How do we rewrite the equation 2x + y = 15 in
the form y = mx + b?
Solution:
2x + y = 15
2x + y + (-2x) = 15 + (-2x)
y = 15 + (-2x)
y = -2x + 15
Given
Addition Property of Equality
Simplification

9. How do we rewrite the equation 3x – 5y = 10
in the form y = mx + b?
Solution:
3x – 5y = 10
3x – 5y + (-3x) = 10 + (-3x)
-5y = -3x + 10
−
1
5
−5𝑦 = −
1
5
−3𝑥 + 10
𝑦 =
3
5
𝑥 − 2
Given
Addition Property of Equality
Simplification
Multiplication Property of
Equality
Simplification|

10. How do we rewrite the equation y = -5x + 16 in
the form Ax + By = C?
Solution:
y = -5x + 16
5x + y = -5x + 16 + 5x
5x + y = 16
Given
Addition Property of Equality
Simplification

11. How do we rewrite the equation y =
𝟏
𝟐
x + 3 in
the form Ax + By = C?
Solution:
y =
𝟏
𝟐
𝒙 + 3
2𝑦 = 2
𝟏
𝟐
𝑥 + 3
2y = x + 6
2y + (-x) = x + 6 + (-x)
-x + 2y = 6
(-1)(-x + 2y) = (-1)(6)
x – 2y = -6
Given
Multiplication Property of Equality
Simplification
Addition Property of Equality
Simplification
Multiplication Property of Equality
Simplification

13. Graphing Linear Equations
A first-degree polynomial equation in two variables is said to
be a linear equation. The graph of linear equation is a line.
Graphing linear equations can be done using any of the three
methods.
1. Using any two points on the line
2. Using 𝑥 and 𝑦- intercepts
3. Using the slope and a point
Page 184 on your Mathematics Book

14. y
x
(0, 3)
(-4, 0)
y
x
(0, 5)
(3, 0)
The graph of a linear equation is
always a ____.
line

15. Recall that, the Cartesian Plane is
consist of two perpendicular number lines
intersecting at the origin. The position,
direction and distance of all points in the
plane relative to the origin are given by its
coordinates, the ordered pair (x, y). The 𝑥
- coordinate or abscissa of a point is its
horizontal distance from the origin. The 𝑦-
coordinate or the ordinate of a point is
its vertical distance from the origin.
Hence, divided the plane into four
regions, Quadrant I, II, III, and IV. Then,
we can describe any point of the plane
using ordered pair of numbers. The ordered pair (4, 3) is located
at quadrant I as it is shown
above.

16. 1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
-9 -8 -7 -6 -5 -4 -3 -2 -1
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
(8,6)
(2,-3)
(-4,-10)
(-3,2)
C
A
D
B

17. Graph a linear equation
given any two points
Example: (-5, 1) and (0, -4)
1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
-9 -8 -7 -6 -5 -4 -3 -2 -1
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
Given two point (x1, y1) and
(x2, y2), just simply plot the
given on the Cartesian
coordinate plane then
connect the two points.

18. Graph a linear equation
given any two points
Example: (-5, 1) and (0, -4)
1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
-9 -8 -7 -6 -5 -4 -3 -2 -1
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
(-5, 1)
(0, 4)

19. Graph a linear equation
given any two points
Example: (1, 6) and (3, 1)
1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
-9 -8 -7 -6 -5 -4 -3 -2 -1
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
(1, 6)
(3, 1)

20. Using Slope and One
Point
Example:
m = 3, through (1,3)
1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
-9 -8 -7 -6 -5 -4 -3 -2 -1
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1. Plot the given point (1, 3)
2. Use the slope formula 𝑚 =
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
to
identify the rise and the run.
The slope of the line is 3 which is
equal to
3
1
.
3. Starting at the given point (1, 3),
count out the rise (3 units up) and
run (1 unit to the right) to mark the
second point. (Note that the slope
is positive)
4. Draw a line passing the points.
(1,3)
(2,6)

22. Describing the Graph of a
Linear Equation
A line can be described by its slope. The
slope of a line is a number that measures its
"steepness", usually denoted by the letter 𝑚.
It is the change in 𝑦 for a unit change in 𝑥
along the line.
Page 182 on your Mathematics Book

23. Describing the Graph of a
Linear Equation
In graphing linear equation using the slope and intercept of the
equation, you simply identify the slope and the y-intercept given an
equation. If the equation is in the general form, ax + by = c, the slope
is −
𝒂
𝒃
and the y-intercept is
𝒄
𝒃
. If the equation is in slope-intercept form,
y = mx + b, m is the slope and b is the y-intercept.

24. Steps:
1. Identify the slope and y-intercept.
2. Plot the y-intercept (0, y) on the Cartesian plane.
3. From the point of the y-intercept, plot the slope.
Note that the slope is the ratio of change in y and change in x,
or
Slope (m) =
ࢉࢎࢇ࢔ࢍࢋ ࢏࢔ 𝒚
ࢉࢎࢇ࢔ࢍࢋ ࢏࢔ 𝒙
=
࢜ࢋ࢚࢘࢏ࢉࢇ࢒ ࢉࢎࢇ࢔ࢍࢋ
ࢎ࢕࢘࢏ࢠ࢕࢔࢚ࢇ࢒ ࢉࢎࢇ࢔ࢍࢋ
=
rise
run
4. Then connect the two points.

25. Taken from Mathematics 8 Learners Module

28. Describe the graph of the following
linear equations in terms of its
intercepts and slope.
Example: 3x + y = 3
a = 3, b = 1, c = 3
Slope (m) = −
𝒂
𝒃
= −
𝟑
𝟏
=
𝒓𝒊𝒔𝒆(𝒅𝒐𝒘𝒏)
𝒓𝒖𝒏
y-intercept (b) =
𝒄
𝒃
=
𝟑
𝟏
= 𝟑 𝒐𝒓 (𝟎, 𝟑)
1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
-9 -8 -7 -6 -5 -4 -3 -2 -1
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
(0, 3)
(1, 0)
Description: The graph is decreasing
from left to right.

29. Describe the graph of the following
linear equations in terms of its
intercepts and slope.
Example: x – 2y = 8
a = 1, b = -2, c = 8
Slope (m) = −
𝟏
−𝟐
=
𝟏
𝟐
=
𝒓𝒊𝒔𝒆(𝒖𝒑)
𝒓𝒖𝒏
y-intercept (b) =
𝒄
𝒃
=
𝟖
−𝟐
= −𝟒 𝒐𝒓 (𝟎, −𝟒)
1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
-9 -8 -7 -6 -5 -4 -3 -2 -1
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
(0, 3)
(1, 0)
Description: The graph is increasing
from left to right.