Download 3.1 Solving Equations Using Addition and Subtraction

3-3A Linear Functions
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
In Chapter 2 you solved linear equations. In a linear
equation the exponent of the variable is one.
x1 12  40
In this lesson you will graph linear equations in two
variables. In a linear equation with two variables
the exponent of the variables is one.
1
1
xy 4
A linear equation in x and y is an equation that can be
written in the form Ax + By = C.
2x  3y  12
3x  y  8
x  4y  6
xy 7
This is the
standard form
of a linear
equation!
The A is the
coefficient of the
x term while the B
is the coefficient
of the y term! C
represents a
constant.
In standard form A > 0, and A and B are not both zero and
A, B, and C are integers with a greatest common factor of
1.
Explain why the following equations are NOT
considered to be in standard form.
2x  12  3y x and y need to be on the left side of equal sign.
y  3x  8
x term comes before the y term.
 2x  4 y  6 “A” needs to be greater than or equal to one.
2
xy 7
3
2x  4 y  8
“A” is not an integer.
GCF of A, B, and C is not 1.
Determine whether y = 5 – 2x is a linear equation. If so,
write the equation in standard form.
Determine if linear equation.
Write in standard form.
Add 2x to each side.
Simplify.
Since the x
and y terms
are to the
first degree, it
is a linear
equation.
y  5  2x
 2x
 2x
2x  y  5
Determine whether each equation is a linear equation.
If so, write the equation in standard form.
Example 1
5x + 3y = z + 2
no
Since there
are 3
variables it is
not a linear
equation.
Example 2
3
x  y8
4
4  3 x   y  84 
4 
3x  4 y  32
 4y  4y
Example 3
2xy  5y  6
3x  4 y  32
Since the
term 2xy has
two variables,
the equation
is not a linear
equation.
no
Graphing Linear Equations
All graphs must be completed neatly on graph paper.
Show your support work next to the graph on the
graph paper or it can be completed on regular
notebook paper.
The graph of an equation in x and y is the set of all points
(x,y) that are solutions of the equation.
The graph of a linear equation is a straight line.
The root word for linear is line.
y
x
A default increment of one will be used if no increment values are on
the graph. Label your graph if you are using an increment other than
one. Draw and label the x and y axis on your graph.
Graphing A Linear Equation
Solve the equation for y, if necessary.
Choose at least three values of x and make a table of values.
Choose values that will produce integer coordinates.
Plot the points from the table of values and draw a line
through the points. All points on the line are solutions of
the equation.
x
Draw the line
through the
entire graph.
Use
arrowheads!
y
Graph the equation –2x + y = –3.
Write equation.
Solve the equation for y.
Reminder: place the variable term
before the constant.
 2x  y  3
 2x
 2x
y  2x  3
Choose any three values for x, and make a table of values.
It is a good idea to use positive and negative values.
Choose values that will produce integer coordinates.
The x is the independent variable and the y is the
dependent variable.
The best values
to choose when
the coefficient
of x is an
integer is -1, 0,
1
Graph –2x + y = –3.
Create a table of values.
Plot the points and draw
a line through them.
x
1
0
1
y = 2x – 3
y
(x,y)
y  2 1  3
 2   3
 5
 1,5
y  20  3
 3
(0,3)
y  2(1)  3
 2  (3)
(1,1)
 1
The x is the independent variable and
the y is the dependent variable.
•
•
•
x
Graph the equation –x + 4y = –4.
Write equation.
Solve the equation for y.
Reminder: place the variable term
before the constant.
 x  4 y  4
x
x
4y  x  4
4 4 4
1
y  x 1
4
Choose any three values for x, and make a table of values.
It is a good
use to
positive and negative values.
The idea
best to
values
Choose choose
valuesfor
that
will produce integer coordinates.
fractional
coefficient of x are
multiples of the
denominator and
zero.
Efficient values to choose for x.
It is a good idea to use positive and negative values.
Choose values that will produce integer coordinates.
When the equation is solved for y and the coefficient of
the x term is a fraction, choose multiples of the
denominator.
2
y  x  1  5, 0, 5
5
When the equation is solved for y and the coefficient of
the x term is an integer, choose -1, 0, and 1.
y  5x  1
Graph –x + 4y = –4.
y
Create a table of values.
Plot the points and draw
a line through them.
x y  1 x  1 (x,y)
4
 4 y  1  4  1
0
4
4
 1  1
 4,2
 2
1
y  0   1
4
(0,1)
 1
1
y  (4)  1
4
1 1
(4,0)
0
•
•
•
x
Example 4 Graph the equation -½x + y = –3.
Example 5 Graph the equation x – 3y = –9.
Example 6 Graph 2x – 4y = –16.
Write equation.
Solve the equation for y.
Choose any three values for x, and make a table of
values. Choose values that will produce integer
coordinates.
Plot the points from the table of values and draw a
line through the points. All points on the line are
solutions of the equation.
Example 4 Graph the equation -½x + y = –3.
1
 x  y  3
2
1
1
 x
 x
2
2
1
y  x 3
2
Since you can choose any values for x,
use values which are multiples of 2 to
avoid graphing fractional values.
Example 4 Graph the equation -½x + y = –3.
x y  1 x 3
2
(x,y)
 2 y  1   2  3
2
 1  3
 4
0 y  1 0   3
2
 3
( 2,4)
0,3
2 y  1 2  3
•
•
•
2
 1 3
 2
(2,-2)
y
x
Example 5 Graph the equation x – 3y = –9.
x  3y  9
x
x
 3y  x  9
3 3 3
1
y  x  3
3
1
y  x 3
3
Since you can choose any values for x,
use values which are multiples of 3 to
avoid graphing fractional values.
Example 5 Graph the equation x – 3y = –9.
x y  1 x 3
3
(x,y)
 3 y  1  3  3
3
 1  3
2
(3,2)
0 y  1 0  3
3
3
•
•
•
0,3
3 y  1 3  3
3
13
4
(3,4)
y
x
Example 6 Graph 2x – 4y = –16.
2x  4y  16
 2x
 2x
 4y  2x  16
4 4
4
1
y  x  4
2
y
1
x4
2
Since you can choose any values for x,
use values which are multiples of 2 to
avoid graphing fractional values.
Example 6 Graph 2x – 4y = –16.
x
y
1
x4
2
 2 y  1  2  4
2
 1  4
(x,y)
(2,3)
3
0 y  1 0  4
2
0,4
4
2 y  1 2  4
2
(2,5)
5
•
•
•
x
y
3-A4 Pages 159–161 #12–17,25,28–30,64–70.