Transcript write a linear equation in standard form. - PMS-Math

Students will be able to write a linear equation in
standard form.
Warm-Up
Write an equation in point-slope form of the line that
passes through the given points.
1. (1, 4), (6, –1)
ANSWER
y – 4 = –(x – 1) or y + 1 = –(x – 6)
2. ( –1, –2), (2, 7)
ANSWER
y + 2 = 3(x + 1) or y – 7 = 3(x – 2)
3. A store rents 3 DVDs for $5, plus $3 for each
additional DVD. Find the cost of renting 20 DVDs.
ANSWER
$56
 Review Homework
Students will be able to write a linear equation in
standard form.
Students will be able to write a linear equation in
Daily Homework Quiz standard form.
1.
2.
Write an equation in point-slope form of the line
that passes through (6, – 4) and has slope 2.
y + 4 = –2(x –6)
ANSWER
Write an equation in point-slope form of the line
that passes through (–1, –6) and (3,10).
ANSWER
3.
y + 6 = 4(x + 1) or y –10 = 4(x–3)
A travel company offers guided rafting trips for
$875 for the first three days and $235 for each
additional day. Write an equation that gives the
total cost (in dollars) of a rafting trip as a function
of the length of the trip. Find the cost for a 7-day
trip.
C = 235t + 170, where C is total cost and t is
ANSWER
time (in days); $1815
Students will be able to write a linear equation in
standard form.
Methods to Represent Linear Functions
 Slope Intercept Form: y = mx + b
 Point-Slope Form: y – y1 = m(x – x1)
 m = slope
 (x1, y1) = point on the line
 Standard Form: Ax + By = C
 A, B, and C are real numbers.
 Useful to model real life situations….
 Not useful for graphing
EXAMPLE 1
Students will be able to write a linear equation in
standard form.
Write equations
in standard form
Write these equations in standard form.
y = 2x – 9
2x – y = 9
y = 6 - 5x
5x + y = 6
y=9+x
x – y = -9
y + 1 = 3(x + 1)
-3x + y = 2
y – 2 = 5(x – 11)
-5x + y = -53
EXAMPLE 2
Students will be able to write a linear equation in
Write an equation
from a graph
standard form.
Write an equation in standard form of the line shown.
SOLUTION
STEP 1
Calculate the slope.
1 – (–2)
3
m=
= –1 = – 3
1 –2
STEP 2
Write an equation in point-slope form. Use (1, 1).
y – y1 = m(x – x1)
y – 1 = – 3(x – 1)
Write point-slope form.
Substitute 1 for y1, 23 for m
and 1 for x1.
EXAMPLE 2
Students will be able to write a linear equation in
Write an equation
from a graph
standard form.
STEP 3
Rewrite the equation in standard form.
3x + y = 4
Simplify. Collect variable
terms on one side,
constants on the other.
Students will be able to write a linear equation in
from 1a and
graph
standard
form.
EXAMPLE
2 Write an equation
for Examples
2
GUIDED PRACTICE
2
Write an equation in standard form of the line
through (3,–1) and (2, – 3).
SOLUTION
STEP 1
Calculate the slope.
m=
–3–(–1)
–2
= –1 = 2
2 –3
STEP 2
Write an equation in point-slope form. Use (3, –1).
y – y1 = m(x – x1)
y + 1 = 2(x – 3)
Write point-slope form.
Substitute 3 for x1, –1 for y1
and 2 for m.
Students will be able to write a linear equation in
from 1a and
graph
standard
form.
EXAMPLE
2 Write an equation
for Examples
2
GUIDED PRACTICE
STEP 3
Rewrite the equation in standard form.
– 2x + y = –7
Simplify. Collect variable
terms on one side,
constants on the other.
EXAMPLE 3
Students will be able to write a linear equation in
standard
form. of a line
Write an
equation
Write an equation of the specified line.
a.
Blue line
b.
Red line
SOLUTION
a.
b.
The y-coordinate of the given point on the blue
line is –4. This means that all points on the line
have a y-coordinate of –4. An equation of the
line is y = –4.
The x-coordinate of the given point on the red
line is 4. This means that all points on the line
have an x-coordinate of 4. An equation of the
line is x = 4.
EXAMPLE 4
3
Students will be able to write a linear equation in
Complete
an equation
in standard form
standard
form.
Find the missing coefficient in the equation of the line
shown. Write the completed equation.
SOLUTION
STEP 1
Find the value of A. Substitute the
coordinates of the given point for x and y in
the equation. Solve for A.
Ax + 3y = 2
A(–1) + 3(0) = 2
–A = 2
A=–2
Write equation.
Substitute – 1 for x and 0 for y.
Simplify.
Divide by – 1.
EXAMPLE 4
Students will be able to write a linear equation in
Complete
an equation
in standard form
standard
form.
STEP 2
Complete the equation.
– 2x + 3y = 2
Substitute – 2 for A.
GUIDED PRACTICE
Students will be able to write a linear equation in
standard
form. 3 and 4
for Examples
Write equations of the horizontal and vertical lines that
pass through the given point.
3.
(–8, –9)
SOLUTION
STEP 1
The y-coordinate of the given point is–9. This means
that all points on the line have a y-coordinate of –9 .
An equation of the line is y = –9.
STEP 2
The x-coordinate of the given point is –8. This means
that all points on the line have an x-coordinate of –8.
An equation of the line is x = –8.
GUIDED PRACTICE
Students will be able to write a linear equation in
standard
form. 3 and 4
for Examples
Write an equation of the horizontal and vertical lines
that pass through the given point.
4. (13, –5)
SOLUTION
STEP 1
The y-coordinate of the given point is –5. This means
that all points on the line have a y-coordinate of –5.
An equation of the line is y = –5.
STEP 2
The x-coordinate of the given point is 13. This means
that all points on the line have an x-coordinate of 13.
An equation of the line is x = 13.
Students will be able to write a linear equation in
an
equation
standard
form
standard
form. of
EXAMPLE
4
3 Complete
for
Examples
3 in
and
4
Write an
equation
a line
GUIDED PRACTICE
Find the missing coefficient in the equation of the
line that passes through the given point. Write the
completed equation.
5.
–4x+By = 7, (–1,1)
SOLUTION
STEP 1
Find the value of B. Substitute the coordinates of the
given point for x and y in the equation. Solve for B.
–4x + By = 7
–4(–1) + B(1) = 7
B=3
Write equation.
Substitute –1 for x and 1 for y.
Simplify.
Students will be able to write a linear equation in
anExamples
equation
form
standard
form. 3 in
EXAMPLE
4 Complete
for
andstandard
4
GUIDED PRACTICE
STEP 2
Complete the equation.
– 4x + 3y = 7
Substitute 3 for B.
Students will be able to write a linear equation in
standard form.
Real Life Example
 Standard Form: Ax + By = C
 Example
 You have $50 to spend at a used book store.
 Paperbacks (x): $1, Hardcovers (y) $4
1x + 4y = 50
If I want to buy 7 hardcover books, how many paperback
books could I buy?
1x + 4(7) = 50
-28
-28
x = 22
Students will be able to write a linear equation in
an
equation
standard
form
standard
form. of
EXAMPLE
4
3 Complete
for
Examples
3 in
and
4
Write an
equation
a line
GUIDED PRACTICE
Find the missing coefficient in the equation of the line
that passes through the given point. Write the
completed equation.
6. Ax+y = –3, (2, 11)
SOLUTION
STEP 1
Find the value of A. Substitute the
coordinates of the given point for x and y in
the equation. Solve for A.
Write equation.
Ax + y = –3
Substitute 2 for x and 11 for y.
A(2) + 11 = –3
Simplify.
2A= –14
Divide each side by 2.
A= –7
Students will be able to write a linear equation in
anExamples
equation
form
standard
form. 3 in
EXAMPLE
4 Complete
for
andstandard
4
GUIDED PRACTICE
STEP 2
Complete the equation.
– 7x +y = –3
Substitute –7 for A.
EXAMPLE 5
Students will be able to write a linear equation in
Solve a multi-step
problem
standard form.
Library
Your class is taking a trip to the public library. You can
travel in small and large vans. A small van holds 8
people and a large van holds 12 people. Your class
could fill 15 small vans and 2 large vans.
a. Write an equation in standard form that models
the possible combinations of small vans and
large vans that your class could fill.
b. Graph the equation from part (a).
c. List several possible combinations.
EXAMPLE 5
Students will be able to write a linear equation in
Solve a multi-step
problem
standard form.
SOLUTION
a. Write a verbal model. Then write an equation.
8
s
+
12
p
l
=
Because your class could fill 15 small vans and 2
large vans, use (15, 2) as the s- and l-values to
substitute in the equation 8s + 12l = p to find the
value of p.
8(15) + 12(2) = p
Substitute 15 for s and 2 for l.
Simplify.
144 = p
Substitute 144 for p in the equation 8s + 12l = p.
EXAMPLE 5
Students will be able to write a linear equation in
Solve a multi-step
problem
standard form.
ANSWER
The equation 8s + 12l = 144 models the possible
combinations.
b.
Find the intercepts of the graph.
Substitute 0 for s.
8(0) + 12l = 144
l = 12
Substitute 0 for l.
8s + 12(0) = 144
s = 18
EXAMPLE 5
Students will be able to write a linear equation in
Solve a multi-step
problem
standard form.
Plot the points (0, 12) and (18, 0). Connect them
with a line segment. For this problem only
nonnegative whole-number values of s and l make
sense.
c. The graph passes through (0, 12), (6, 8),(12, 4), and
(18, 0). So, four possible combinations are 0 small
and 12 large, 6 small and 8 large, 12 small and 4
large, 18 small and 0 large.
Students will be able to write a linear equation in
problem
standard
form.
EXAMPLE
5 Solve
for Example
5
GUIDED PRACTICE
Solve aa multi-step
multi-step
problem
EXAMPLE 5
7. WHAT IF? In Example 5, suppose that 8 students
decide not to go on the class trip. Write an equation
that models the possible combinations of small and
large vans that your class could fill. List several
possible combinations.
Students will be able to write a linear equation in
problem
standard
form.
EXAMPLE
5 Solve
for Example
5
GUIDED PRACTICE
Solve aa multi-step
multi-step
problem
EXAMPLE 5
SOLUTION
STEP 1 Write a verbal model. Then write an equation.
8
s
+
12
p
l
=
8 students decide not to go on the class trip, so the
class could fill 14 small vans and 2 large vans. Because
your class could fill 14 small vans and 2 large vans, use
(14, 2) as the s- and l-values to substitute in the equation
8s + 12l = p to find the value of p.
8(14) + 12(2) = p
Substitute 14 for s and 2 for l.
Simplify.
136 = p
Substitute 136 for p in the equation 8s + 12l = p.
Students will be able to write a linear equation in
problem
standard
form.
EXAMPLE
5 Solve a multi-step
for Example
5
GUIDED PRACTICE
ANSWER
The equation 8s + 12l = 136 models the possible
combinations.
STEP 2 Find the intercepts of the graph.
Substitute 0 for s.
8(0) + 12l = 136
4
l = 11
12
Substitute 0 for l.
8s + 12(0) = 136
s = 17
GUIDED PRACTICE
Students will be able to write a linear equation in
standard
form. 5
for Example
Plot the point(0,11 4 )and(17, 0).connect them with
12
a line segment. For this problem only negative
whole-number values of s and l make sense.
STEP 3
The graph passes through (17, 0), (14, 2), (11, 4),
(8, 6), (5, 8) and (2, 10). So, several combinations
are 17 small, 0 large; 14 small 2 large; 11 small,
4 large; 18 small, 6 large; 5 small, 8 large; 2 small,
10 large.