Download Section 3.1: Introduction to Linear Equations in 2 Variables Section

Remember to read the textbook before attempting to do your homework.
Section 3.1: Introduction to Linear Equations in 2 Variables
Section 3.2: Graphing by Plotting Points and Finding Intercepts
y
Rectangular Coordinate System
aka Cartesian coordinate system, aka xy-plane
The x-axis (horizontal axis) and y-axis (vertical axis) divide
the plane into four quadrants (regions):
The graph of an ordered pair, (x, y), is a point where
the 1st number is called the x-coordinate,
the 2nd number is called the y-coordinate.
Quadrant 2
(–,+)
Origin
Things to note:
Quadrant 3
The origin is the point where the x-and y-axis meet.
(–,–)
The coordinates of the origin are given by (0, 0).
The points on the x-axis or y-axis do not belong to any quadrant.
The order of the numbers is very important, e.g. (2, 5) is not the same as (5, 2)
Example 1:
a. Is (3, 5) a solution of 2x – 3y = –9 ?
Quadrant 1
(+,+)
(0, 0)
x
Quadrant 4
(+,–)
b. Is (5, 3) a solution of 2x – 3y = –9 ?
Example 2: Graph the following set of ordered pairs and
label the coordinates of the points:
y
{(0, 0), (–1, 3), (0, –5), (5, 0), (3.5, 4), (–3, –4)}
Friendly Reminder:
Ordered pairs are written in the form (x, y). To graph
an ordered pair, first start from the origin, and then . . .
x
1. See how many units you should move to the left or
right (this is determined by the x-coordinate)
2. See how many units you should move up or down
(this is determined by the y-coordinate)
Math 60 || Beginning Algebra || Cerritos College
– Pg 1 –
Chapter 3 Lecture Notes by Maria Torres
Standard Form
A linear equation in 2 variables written in the form Ax + By = C, is said to be in standard form,
where A, B, and C are real numbers (A and B not both 0).
In English: Fix the equation so you have the x-term + y-term = constant term.
Also, please make sure that you have integer coefficients.
Question: I know that the graph of an ordered pair is a point, but what does the graph of a linear
equation look like?
Answer: A straight line.
Question: How do you graph of a line?
Answer: Oh that’s easy! Just find 2 ordered pair solutions of the equation, graph them, and then
connect the points.
Question: Is there an easy way to find ordered pair solutions?
Answer: Of course there is! Just ...
1. Solve the equation for y, then
2. Create an xy-table, and then
3. Pick a value for x and find the corresponding value for y. ☺
Ex 3: Graph by plotting at least 3 points for each graph. You may not use a calculator to answer.
a. –6x + 2y = 4
y
x
b. 4x = –3y – 9
y
x
Math 60 || Beginning Algebra || Cerritos College
– Pg 2 –
Chapter 3 Lecture Notes by Maria Torres
Question: Is there a different way to obtain the graph of a line?
Answer: Yes, there is! We can also graph a line by using the x-intercept and the y-intercept.
y
Intercepts of a Line
Since the graph
x-intercept: The point where the line intercepts the x-axis.
To find the x-intercept:
Let y = 0, and then Solve for x. intercepts the
x-axis at 1, then
Note: The x-intercept is of the form (x1, 0)
x-int = (1, 0)
y-intercept: The point where the line intercepts the y-axis.
To find the y-intercept:
Let x = 0, and then Solve for y. Since the graph
intercepts the
Note: The y-intercept is of the form (0, y1)
y-axis at –2, then y-int = (0, –2)
x
Example 4: What is the x-intercept? What is the y-intercept? Use the x-intercept and y-intercept to
graph the line. Label the coordinates of the intercepts.
y
a. –4x + 5y = 20
x
y
b. 14 – 4y = 7x
x
Bored or just looking for more fun? Then work on the attached Xtra Practice Sheet (intercepts)
Math 60 || Beginning Algebra || Cerritos College
– Pg 3 –
Chapter 3 Lecture Notes by Maria Torres
Xtra Practice: Using intercepts to graph linear equations
Sample: Use the intercepts to graph –2x – 3y = 6. On your graph, please label the coordinates of the
intercepts.
Solution:
Goal #1:
Goal #2:
Goal #3:
Find the x-intercept
Find the y-intercept
Use the intercepts to graph the line.
To find the x-intercept:
Let y = 0, then solve for x.
To find the y-intercept:
Let x = 0, then solve for y.
–2x – 3y = 6
–2x – 3y = 6
–2x – 3(0) = 6
–2(0) – 3y = 6
–2x = 6
–3y = 6
x = –3
(–3, 0)
(0, –2)
y = –2
P.S.
x-intercept: (–3, 0)
y-intercept: (0, –2)
Don’t forget to label the intercepts
Directions: What is the x-intercept? What is the y-intercept? Use the x-intercept and y-intercept to
graph the line. Label the coordinates of the intercepts
y
y
x
a. 5x – 6y = –30
Math 60 || Beginning Algebra || Cerritos College
x
b. 8y – 4x = –32
– Pg 4 –
Chapter 3 Lecture Notes by Maria Torres
y
y
x
c. –4 = – 4y + x
x
d. 5x = 35 – 7y
y
y
x
e.
18 = 2y
x
f. –y = 4x + 6
(Solutions will be posted on the Math 60 page)
Math 60 || Beginning Algebra || Cerritos College
– Pg 5 –
Chapter 3 Lecture Notes by Maria Torres
Vertical and Horizontal Lines
The graph of any equation of the form
The graph of any equation of the form
x = x1
y = y1
is a vertical line that intercepts the x-axis at x1.
is a horizontal line that intercepts the y-axis at y1.
Moral of the story: If an equation does not contain BOTH x and y variables, then the graph of
the equation will be either a vertical line or a horizontal line.
Example 5: Graph x = 2
Example 6: Graph y = –3
Psst psst . . .
The graph of x = 2 is a
vertical line that intercepts the
x-axis at positive 2.
Psst psst . . .
the graph of y = –3 is a
horizontal line that intercepts
the y-axis at negative 3.
Example 7: Graph the lines.
a. x + 3 = 0
b. y – 8 = 0
c. 3y + 10 = –2
d. 2x – 14 = –2
(end of Section 3.1/3.2 combo)
Math 60 || Beginning Algebra || Cerritos College
– Pg 6 –
Chapter 3 Lecture Notes by Maria Torres