Download and y-intercepts.

5-6
Slope-Intercept Form
Objectives
Find the two intercepts
Graph a line using intercepts
Rewrite equations in slope-intercept form.
Identify the slope and y-intercept
Graph a line using slope-intercept form.
Solve systems of linear equations by graphing
Holt Algebra 1
5-6
Slope-Intercept Form
The y-intercept is the ycoordinate of the point
where the graph intersects
the y-axis. The x-coordinate
of this point is always 0.
The x-intercept is the xcoordinate of the point
where the graph intersects
the x-axis. The y-coordinate
of this point is always 0.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 1A: Finding Intercepts
Find the x- and y-intercepts.
–3x + 5y = 30
To find the x-intercept,
To find the y-intercept,
replace y with 0 and solve
replace x with 0 and solve
for x. –3x + 5y = 30
for y. –3x + 5y = 30
–3x + 5(0) = 30
–3x – 0 = 30
–3x = 30
x = –10
The x-intercept is (–10, 0).
Holt Algebra 1
–3(0) + 5y = 30
0 + 5y = 30
5y = 30
y=6
The y-intercept is (0,6).
5-6
Slope-Intercept Form
Example 1B
Find the x- and y-intercepts.
4x + 2y = 16
To find the x-intercept,
To find the y-intercept,
replace y with 0 and solve
replace x with 0 and solve
for x.
for y. 4x + 2y = 16
4x + 2y = 16
4x + 2(0) = 16
4x + 0 = 16
4x = 16
x=4
The x-intercept is (4,0).
Holt Algebra 1
4(0) + 2y = 16
0 + 2y = 16
2y = 16
y=8
The y-intercept is (0,8)
Slope-Intercept Form
5-6
Example 2: Sports Application
Trish can run the 200 m dash in 25 s. The
function f(x) = 200 – 8x gives the distance
remaining to be run after x seconds. Graph
this function and find the intercepts. What
does each intercept represent?
Neither time nor distance can be negative, so choose
several nonnegative values for x. Use the function to
generate ordered pairs.
x
f(x) = 200 – 8x
Holt Algebra 1
0
5
10
20
25
200
160
120
40
0
5-6
Slope-Intercept Form
Example 2 Continued
Graph the ordered pairs. Connect
the points with a line.
y-intercept(0, 200). This is the
number of meters Trish has to
run at the start of the race.
x-intercept: (25, 0). This is the
time it takes Trish to finish the
race {remaining distance is 0}.
Holt Algebra 1
5-6
Slope-Intercept Form
Remember, to graph a linear function, you need
to plot only two ordered pairs. It is often
simplest to find the ordered pairs that contain
the intercepts.
Helpful Hint
You can use a third point to check your line. Either
choose a point from your graph and check it in the
equation, or use the equation to generate a point
and check that it is on your graph.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 2
Use intercepts to graph the line –x + 3y = –6
x-intercept:
–x = –6
y-intercept:
3y = –6
Plot (6, 0) and (0, –2).
Connect with a straight
line.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 3: Word Problem
3. An amateur filmmaker has $6000 to make a film
that costs $75/h to produce. The function f(x) =
6000 – 75x gives the amount of money left to
make the film after x hours of production. Graph
this function and find the intercepts. What does
each intercept represent?
x-int.(80,0); number of hours it takes
to spend all the money
y-int.(0, 6000); the amount of money
available.
Holt Algebra 1
5-6
Slope-Intercept Form
Any linear equation can be written in slope-intercept
form by solving for y and simplifying. In this form,
you can immediately see the slope and y-intercept.
Also, you can quickly graph a line when the equation
is written in slope-intercept form.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 4: Using Slope-Intercept Form to Graph
Write the equation in slope-intercept form.
Then graph the line described by the equation.
y = 3x – 1
y = 3x – 1 is in the form y = mx + b
slope: m = 3 =
•
y-intercept: (0,–1)
Step 1 Plot (0, –1).
Step 2 Count 3 units up and 1
unit right and plot another point.
Step 3 Draw the line connecting
the two points.
Holt Algebra 1
•
5-6
Slope-Intercept Form
Example 5: Using Slope-Intercept Form to Graph
Write the equation in slope-intercept form.
Then graph the line described by the equation.
2y + 3x = 6
Step 1 Write the equation in slope-intercept form
by solving for y.
2y + 3x = 6
–3x –3x
2y = –3x + 6
Subtract 3x from both sides.
Since y is multiplied by 2,
divide both sides by 2.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 5 Continued
Write the equation in slope-intercept form.
Then graph the line described by the equation.
Step 2 Graph the line.
is in the form
y = mx + b.
slope: m =
y-intercept: (0, 3)
Plot (0, 3).
•
•
• Count 3 units down and 2 units right and plot
another point.
• Draw the line connecting the two points.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 6
Write the equation in slope-intercept form.
Then graph the line described by the equation.
6x + 2y = 10
Step 1 Write the equation in slope intercept form
by solving for y.
6x + 2y = 10
–6x
–6x
2y = –6x + 10
Subtract 6x from both sides.
Since y is multiplied by 2,
divide both sides by 2.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 6 Continued
Write the equation in slope-intercept form.
Then graph the line described by the equation.
Step 2 Graph the line.
•
y = –3x + 5 is in the form
y = mx + b.
slope: m =
•
y-intercept: (0,5)
• Plot (0, 5).
• Count 3 units down and 1 unit right and plot
another point.
• Draw the line connecting the two points.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 4: Application
A closet organizer charges a $100 initial
consultation fee plus $30 per hour. The cost
as a function of the number of hours worked
is graphed below.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 4 Continued
A closet organizer charges $100 initial
consultation fee plus $30 per hour. The cost
as a function of the number of hours worked
is graphed below.
b. Identify the slope and y-intercept and describe
their meanings.
The y-intercept is (0, 100). This is the cost for 0 hours, or the
initial fee of $100. The slope is 30. This is the rate of change
of the cost: $30 per hour.
c. Find the cost if the organizer works 12 hrs.
y = 30x + 100
Substitute 12 for x in the
= 30(12) + 100 = 460 equation
The cost of the organizer for 12 hours is $460.
Holt Algebra 1
5-6
Slope-Intercept Form
A system of linear equations is a set of two or
more linear equations containing two or more
variables. A solution of a system of linear
equations with two variables is an ordered pair
that satisfies each equation in the system. So, if an
ordered pair is a solution, it will make both
equations true.
Holt Algebra 1
5-6
Slope-Intercept Form
All solutions of a linear equation are on its graph.
To find a solution of a system of linear equations,
you need a point that each line has in common. In
other words, you need their point of intersection.
y = 2x – 1
y = –x + 5
The point (2, 3) is where the
two lines intersect and is a
solution of both equations,
so (2, 3) is the solution of
the systems.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 5
Solve the system by graphing. Check your answer.
y = –2x – 1
y=x+5
Graph the system.
The solution appears to be (–2, 3).
y=x+5
y = –2x – 1
Check Substitute (–2, 3)
into the system.
y = –2x – 1
y=x+5
3
3
–2(–2) – 1
4 –1
3
3
(–2, 3) is the solution of the system.
Holt Algebra 1
3 –2 + 5
3 3
5-6
Slope-Intercept Form
Practice:
Write each equation in slope-intercept form.
Then graph the line described by the equation.
1. 6x + 2y = 10
2. x – y = 6
y=x–6
y = –3x + 5
Holt Algebra 1
5-6
Slope-Intercept Form
Practice: Systems
Solve the system by graphing.
3.
y + 2x = 9
(2, 5)
y = 4x – 3
4. Joy has 5 collectable stamps and will buy 2
more each month. Ronald has 25 collectable
stamps and will sell 3 each month. After how
many months will they have the same number
of stamps? 4 months How many will that be?
13 stamps
Holt Algebra 1