Download Alg 1 - Ch 4.2 Graphing Linear Equations

Steps to Solve Word Problems
1.Read the problem carefully.
2.Get rid of clutter
3.Identify key variables
(unknowns).
4.Eliminate unneeded variables.
5.Use the text of the problem to
write equations.
6.Solve the equation.
7.Find the remaining variables.
Example of Word Problem
If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70,
how much does each apple and each orange cost? There are no quantity
discounts.
What do we know?
If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70,
how much does each apple and each orange cost? There are no quantity
discounts.
Step 1: Now you need to identify the
unknowns. An unknown is a quantity that your
problem requires you to find out, or the
quantity that is necessary to find out in order to
obtain the solution. What are the unknowns in
this problem?
The price of 1 apple and 1 orange
What do we know?
If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70,
how much does each apple and each orange cost? There are no quantity
discounts.
Step 2: Now you need to find out the
equation needed to solve the problem. The
problem basically gives you the equations,
you just need to write them in
mathematical form.
4x + 2y = 1,
2x + 3y = 0.70
What do we know?
If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70,
how much does each apple and each orange cost? There are no quantity
discounts.
Step 2: Now you need to find out the
equation needed to solve the problem. The
problem basically gives you the equations,
you just need to write them in
mathematical form.
4x + 2y = 1,
2x + 3y = 0.70
Now you Try
The senior class at your high school has it
prom at a banquet facility. The banquet
facility $15.95 per person for a dinner buffet
and $400 to rent the banquet hall for an
evening. The class paid the banquet facility
a total of $2633 for the dinner buffet and
use of the banquet hall. How many people
attended the prom?
140 people
Algebra 1
Ch 4.2 – Graphing Linear
Equations
Objective


Students will graph linear equations using
a table.
Students will graph horizontal and vertical
lines
Vocabulary



A linear equation in two variables is an
equation in which the variables appear in
separate terms and neither variable contains an
exponent other than 1.
The solution to linear equations are ordered
pairs which makes the equation true.
The graph of an equation in x and y is the set
of all points (x, y) that are solutions of the
equation.
Example #1

This is an example of a linear equation
y=x+8


All linear equations are functions. That is
the value of y (output) is determined by
the value of x (input)
The linear equation in two variables can
also be called the rule.
Solution
Any ordered pair of numbers
that makes a linear equation
true.
(9,0) IS ONE SOLUTION
FOR Y = X - 9
Examples of Solutions

2y + x = 4

Is (-2,3) a solutions to this equation?
Solution!
Examples of solutions

Graphing



Since the results of a linear equations can be
expressed as ordered pairs, the linear equation
can be graphed.
When a linear equation is graphed, all points on
the line represent the solution set of the linear
equation.
There are a number of ways to find the solution
to a linear equation…for today’s lesson we will
look at creating a table of the solutions…
Tables

To create a table of solutions to a linear
equation do the following:
1.
2.
3.
4.

Choose a minimum of 3 values for x
Substitute the values of x into the linear
equation
Simplify to find the value of y
Write the solutions as ordered pairs
Let’s look at an example…
Example
#2
1. Choose a minimum
y = 2x – 1
of 3 values for x
x
0
y = 2x – 1
y = 2(0) – 1
y
-1
(x, y)
(0, - 1)
1
y = 2(1) – 1
1
(1, 1)
2
y = 2(2) – 1
3
(2, 3)
2. Substitute the value
of x into the equation
3. Simplify to
determine the
value of y
4. Write as an
ordered pair
Graphing



Use the ordered pair from the table to
graph the linear equation.
Again…when graphing the result should
be a straight line…
Any point (ordered pair) on that line will
be a solution to the linear equation…
y
(2,3)
x
(1,1)
(0, -1)
(x, y)
(0, - 1)
(1, 1)
(2, 3)
Linear Equations



All linear equations can be written in the
form:
Ax + By = C
This form is called the standard form of an
equation.
At this level you are required to know and
be able to manipulate this form of an
equation
Standard Form

Ax + By = C
In the standard form of an equation:




A is the coefficient of x
B is the coefficient of y
C represents the constant
We talked about coefficients and constants
in a previous lesson
Example #3


The equation 3x – 4y = 12 is an example of an
equation written in standard form.
As we have done in a previous lesson, we can
write the equation in function form by
transforming the equation as follows:
3x – 4y = 12
-3x
Standard Form
-3x
– 4y = -3x + 12
–4
–4
y=-¾x+3
Function Form
Horizontal Lines



In the standard form of an equation Ax + By
= C, When A=0 the equation reduces to By
= C and the graph will be a horizontal line.
We often see this illustrated as the equation
y = b.
In this instance, the equation has no x-value
and the y-value is always the same number
so that when the y-value is graphed a
horizontal line is produced.
Example #4 – Horizontal Line



Graph the equation y=2
In this instance there is no x-value. All
the y-values = 2
To plot this line, starting at 0, go up 2
spaces on the y-axis and draw a horizontal
line (as shown in the next slide)
Example #4 (Continued)
y=2
y
y=2
x
Comments



Notice that when you graph the line, the
line is perpendicular to the y-axis.
A common error that students make when
graphing an equation like y=2 is that they
draw the line parallel to the y-axis. That is
incorrect!
A way to avoid this error is to actually plot
the point before you draw the line.
Vertical Lines



In the standard form of an equation Ax + By
= C, When B=0 the equation reduces to Ax
= C and the graph will be a vertical line.
We often see this illustrated as the equation
x=a
In this instance, the equation has no y-value
and the x-value is always the same number
so that when the x-value is graphed a vertical
line is produced
Example #5 – Vertical Line



Graph the equation x = -3
In this instance there is no y-value. All
the x-values = -3
To plot this line, starting at 0, go 3 spaces
to the left on the x-axis and draw a
vertical line (as shown in the next slide)
Example #5(Continued)
x=-3
y
x = -3
x
Comments



Notice that when you graph the line, the
line is perpendicular to the x-axis.
A common error that students make when
graphing an equation like x=-3 is that they
draw the line parallel to the x-axis. That is
incorrect!
A way to avoid this error is to actually plot
the point before you draw the line.
Your Turn

1.
2.
3.
4.
Find 3 different ordered pairs that are
the solutions to the equation
y = 3x – 5
y = -2x – 6
y = ½ (4 – 2x)
y = 4( ½ x – 1)
Your Turn

5.
6.

7.
8.
9.
10.
Rewrite the equation in function form
2x + 3y = 6
5x + 5y = 19
Create a table of values & graph the
linear equation
y = -x + 4
y= -(3 – x)
x=9
y = -1
Your Turn Solutions
1.
2.
3.
4.
5.
6.
(-1,-8), (0,-5),(1, -2)
(-1,-4),(0,-6),(1,-8)
(-1,3),(0,2),(1,1)
(-1,-6),(0,-4),(1,-2)
y = -2/3x + 2
y = -x + 19/5

7.
8.
9.
10.
You should have a table with a
minimum of 3 values. When
plotting the line the following
should be true:
Your graph should cross
the y-axis at +4
Your graph should cross
the y-axis at -3
You should have a vertical
line at the point x = 9
You should have a
horizontal line at the point
y= -1
Homework
 Section
4.2 pg.219-220
#16, 11-21(ODD), 23-25,
27, 36