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5-1 Linear Equations and Functions
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Warm Up
California Standards
Lesson Presentation
5-1 Linear Equations and Functions
Warm Up
1. Solve 2x – 3y = 12 for y.
2. Evaluate the function f(x) =
0, 5, and 10.
f(–10) = –1
f(–5) = 0
f(0) = 1
f(5) = 2
f(10) = 3
for –10, –5,
5-1 Linear Equations and Functions
California
Standards
6.0 Students graph a linear equation and
compute x- and y- intercepts (e.g. graph
2x + 6y = 4). They are also able to sketch the
region defined by linear inequalities (e.g., they
sketch the region defined by 2x + 6y < 4).
Also covered:
7.0, 17.0, 18.0
5-1 Linear Equations and Functions
Vocabulary
linear equation
linear function
5-1 Linear Equations and Functions
Many stretches on the
German autobahn have a
speed limit of 120 km/h. If
a car travels continuously at
this speed, y = 120x gives
the number of kilometers y
that the car would travel in
x hours.
Notice that the graph is a straight line. An equation
whose graph forms a straight line is a linear equation.
Also notice that this is a function. A function
represented by a linear equation is a linear function.
5-1 Linear Equations and Functions
For any two points, there is exactly one line that
contains them both. This means you need only
two ordered pairs to graph a line. However,
graphing three points is a good way to check that
your line is correct.
5-1 Linear Equations and Functions
Additional Example 1A: Graphing Linear Equations
Graph y = 2x + 1. Tell whether it represents a
function.
Step 1 Choose three values of
x and generate ordered pairs.
x
y = 2x + 1
(x, y)
1
y = 2(1) + 1 = 3
(1, 3)
0
y = 2(0) + 1 = 1
(0, 1)
–1



y = 2(–1) + 1 = –1 (–1, –1)
Step 2 Plot the points and connect them with a
straight line. No vertical line will intersect this graph
more than once. So y = 2x + 1 describes a function.
5-1 Linear Equations and Functions
Helpful Hint
Sometimes solving for y first makes it easier to
generate ordered pairs using values of x. To
review solving for a variable, see Lesson 2-6.
5-1 Linear Equations and Functions
Additional Example 1B: Graphing Linear Equations
Graph 15x + 3y = 9. Tell whether it represents
a function.
Step 1 Solve for y.
15x + 3y = 9
–15x
–15x
3y = –15x + 9
Subtract 15x from both
sides.
Since y is multiplied by 3
divide both sides by 3.
y = –5x + 3
5-1 Linear Equations and Functions
Additional Example 1B Continued
Graph 15x + 3y = 9. Tell whether it represents
a function.
Step 2 Choose three values of

x and generate ordered pairs
x
y = –5x + 3
(x, y)
1
y = –5(1) + 3 = –2
(1, –2)
0
y = –5(0) + 3 = 3
(0, 3)
–1
y = –5(–1) + 3 = 8
(–1, 8)


Step 3 Plot the points and connect them with a
straight line. No vertical line will intersect this graph
more than once. So 15x + 3y = 9 describes a function.
5-1 Linear Equations and Functions
Additional Example 1C: Graphing Linear Equations
Graph x = –2. Tell whether it represents a
function.
Any ordered pair with an xcoordinate of –2 will satisfy this
equation.
Plot several points that have an
x-coordinate of –2 and connect
them with a straight line.



There is a vertical line that intersects this graph more
than once, so x = –2 does not represent a function.
5-1 Linear Equations and Functions
Additional Example 1D: Graphing Linear Equations
Graph y = 8. Tell whether it represents a
function.
Any ordered pair with a
y-coordinate of 8 will satisfy
this equation.
  
Plot several points that have an
y-coordinate of 8 and connect
them with a straight line.
No vertical line will intersect this graph more than
once, so y = 8 represents a function.
5-1 Linear Equations and Functions
Check It Out! Example 1a
Graph y = 4x. Tell whether it represents a
function.
Step 1 Choose three values of
x and generate ordered pairs
x
y = 4x
(x, y)
1
y = 4(1) = 4
(1, 4)
0
y = 4(0) = 0
(0, 0)
–1
y = 4(–1) = –4
(–1, –4)



Step 2 Plot the points and connect them with a
straight line. No vertical line will intersect this graph
more than once. So y = 4x describes a function.
5-1 Linear Equations and Functions
Check It Out! Example 1b
Graph y + x = 7. Tell whether it represents a
function.
Step 1 Solve for y.
y+x=7
–x –x
y = –x + 7
Subtract x from both sides.
5-1 Linear Equations and Functions
Check It Out! Example 1b Continued
Graph y + x = 7. Tell whether it represents a
function.
Step 2 Choose three values of
x and generate ordered pairs
x
y = –x + 7
(x, y)
1
y = –(1) + 7 = 6
(1, 6)
0
y = –(0) + 7 = 7
(0, 7)
–1



y = –(–1) + 7 = 8 (–1, 8)
Step 3 Plot the points and connect them with a
straight line. No vertical line will intersect this graph
more than once. So y + x = 7 describes a function.
5-1 Linear Equations and Functions
Check It Out! Example 1c
Graph
function.
. Tell whether it represents a
Any ordered pair with an xcoordinate of will satisfy
this equation.
Plot several points that have an
x-coordinate of and connect
them with a straight line.



There is a vertical line that intersects this graph
more than once, so x = does not describe a
function.
5-1 Linear Equations and Functions
5-1 Linear Equations and Functions
Additional Example 2A: Determining Whether a
Point is on a Graph
Without graphing, tell whether each point is
on the graph of 2x + 5y = 16.
(3, 2)
Substitute:
2x + 5y = 16
?
2(3) + 5(2) =16
?
6 + 10 =
16
16 = 16 
Since (3, 2) is a solution to 2x + 5y = 16, (3, 2)
is on the graph.
5-1 Linear Equations and Functions
Additional Example 2B: Determining Whether a
Point is on a Graph
Without graphing tell whether each point is on
the graph of 2x + 5y = 16.
(2, 2)
Substitute:
2x + 5y = 16
?
2(2) + 5(2) = 16
?
4 + 10 = 16
14  16 
Since (2, 2) is not a solution to 2x + 5y = 16,
(2, 2) is not on the graph.
5-1 Linear Equations and Functions
Additional Example 2C: Determining Whether a
Point is on a Graph
Without graphing tell whether each point is on
the graph of 2x + 5y = 16.
(8, 0)
Substitute:
2x + 5y = 16
?
2(8) + 5(0) = 16
?
16 + 0 = 16
16 = 16 
Since (8, 0) is a solution to 2x + 5y = 16, (8, 0)
is on the graph.
5-1 Linear Equations and Functions
Check It Out! Example 2a
Without graphing tell whether each point is on
the graph of x – 3y = 12.
(5, 1)
Substitute:
x – 3y = 12
?
5 – 3(1) = 12
?
5 – 3 = 12
2  12 
Since (5, 1) is not a solution to x – 3y = 12,
(5, 1) is not on the graph.
5-1 Linear Equations and Functions
Check It Out! Example 2b
Without graphing tell whether each point is on
the graph of x – 3y = 12.
(0, –4)
Substitute:
x – 3y = 12
?
0 – 3(–4) =
12
?
0 + 12 = 12
12 = 12 
Since (0, –4) is a solution to x – 3y = 12, (0, –4)
is on the graph.
5-1 Linear Equations and Functions
Check It Out! Example 2c
Without graphing tell whether each point is on
the graph of x – 3y = 12.
(1.5, –3.5)
Substitute:
x – 3y = 12
?
1.5 – 3(–3.5) = 12
?
1.5 + 10.5 = 12
12 = 12
Since (1.5, –3.5) is a solution to x – 3y = 12,
(1.5, –3.5) is on the graph.
5-1 Linear Equations and Functions
Linear equations can be written in the standard form
as shown below.
5-1 Linear Equations and Functions
Notice that when a linear equation is written in
standard form.
• x and y both have exponents of 1.
• x and y are not multiplied together.
• x and y do not appear in denominators,
exponents, or radical signs.
5-1 Linear Equations and Functions
5-1 Linear Equations and Functions
Additional Example 3A: Writing Linear Equations in
Standard Form
Write x = 2y + 4 in standard form and give the
values of A, B, and C. Then describe the graph.
x = 2y + 4
–2y –2y
x – 2y = 4
Subtract 2y from both sides.
The equation is in standard form.
A = 1, B = –2, C = 4
The graph is a line that is neither horizontal nor
vertical.
5-1 Linear Equations and Functions
Additional Example 3B: Writing Linear Equations in
Standard Form
Write x = 4 in standard form and give the
values of A, B, and C. Then describe the graph.
x=4
x + 0y = 4
The equation is in standard form.
A = 1, B = 0, C = 4
The graph is a vertical line at x = 4.
5-1 Linear Equations and Functions
Check It Out! Example 3a
Write y = 5x – 9 in standard form and give the
values of A, B, and C. Then describe the graph.
y = 5x – 9
–5x –5x
–5x + y =
–9
Subtract 5x from both sides.
The equation is in standard form.
A = –5, B = 1, C = –9
The graph is a line that is neither horizontal nor
vertical.
5-1 Linear Equations and Functions
Check It Out! Example 3b
Write y = 12 in standard form and give the
values of A, B, and C. Then describe the graph.
y = 12
0x + y = 12
The equation is in standard form.
A = 0, B = 1, C = 12
The graph is a horizontal line at y = 12.
5-1 Linear Equations and Functions
Check It Out! Example 3c
Write x = 2 in standard form and give the
values of A, B, and C. Then describe the graph.
x=2
x + 0y = 2
The equation is in standard form.
A = 1, B = 0, C = 2
The graph is a vertical line at x = 2.
5-1 Linear Equations and Functions
Remember!
• y – x = y + (–x)
• y +(–x) = –x + y
• –x = –1x
• y = 1y
5-1 Linear Equations and Functions
For linear functions whose graphs are not
horizontal, the domain and range are all real
numbers. However, in many real-world situations,
the domain and range must be restricted. For
example, some quantities cannot be negative, such
as distance.
5-1 Linear Equations and Functions
Sometimes domain and range are restricted even
further to a set of points. For example, a quantity
such as number of people can only be whole
numbers. When this happens, the graph is not
actually connected because every point on the
line is not a solution. However, you may see these
graphs shown connected to indicate that the
linear pattern, or trend, continues.
5-1 Linear Equations and Functions
Additional Example 4: Application
The relationship between human years and
dog years is given by the function y = 7x,
where x is the number of human years. Graph
this function and give its domain and range.
Choose several values of x and
make a table of ordered pairs.
x
y = 7x
(x, y)
2
y = 7(2) = 14
(2, 14)
4
y = 7(4) = 28
(4, 28)
6
y = 7(6) = 42
(6, 42)
The ages are
continuous starting
with 0, so the
domain is: {x ≥ 0}
and the range is:
{y ≥ 0}.
5-1 Linear Equations and Functions
Additional Example 4 Continued
Graph the ordered pairs.
Human Years vs. Dog Years

(6, 42)

(4, 28)

(2, 14)
Any point on the
line is a solution
in this situation.
The arrow shows
that the trend
continues.
5-1 Linear Equations and Functions
Check It Out! Example 4
What if…? At another salon, Sue can rent a
station for $10.00 per day plus $3.00 per
manicure. The amount she would pay each day
is given by f(x) = 3x + 10, where x is the
number of manicures. Graph this function and
give its domain and range.
5-1 Linear Equations and Functions
Check It Out! Example 4 Continued
Choose several values of x and make a table of ordered
pairs.
x f(x) = 3x + 10
0 f(0) = 3(0) + 10 = 10
1 f(1) = 3(1) + 10 = 13
2 f(2) = 3(2) + 10 = 16
3 f(3) = 3(3) + 10 = 19
4 f(4) = 3(4) + 10 = 22
5 f(5) = 3(5) + 10 = 25
The number of manicures
must be a whole number, so
the domain is {0, 1, 2, 3, …}.
The range is {10, 13, 16, 19,
…}.
5-1 Linear Equations and Functions
Check It Out! Example 4 Continued
Graph the ordered pairs.
The individual
points are
solutions in this
situation. The
line shows that
the trend
continues.
5-1 Linear Equations and Functions
Lesson Quiz: Part I
Graph each linear equation. Then tell
whether it represents a function.
1. 2y + x = 6
Yes, it is a function.
2. 3y = 12
Yes, it is a function.
5-1 Linear Equations and Functions
Lesson Quiz: Part II
Without graphing, tell whether each point is
on the graph of 6x – 2y = 8.
3. (1, 1) no
4. (3, 5) yes
5. The cost of a can of iced-tea mix at SaveMore
Grocery is $4.75. The function f(x) = 4.75x
gives the cost of x cans of iced-tea mix. Graph
this function and give its domain and range.
D: {0, 1, 2, 3, …}
R: {0, 4.75, 9.50, 14.25, …}