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Do Now 1/11/10
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Text p. 430, #4-20 evens, 30-34 evens
 Text p. 439, #4-24 even, #32, #36
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Chapter 7 “Systems of Equations and
Inequalities”
Chapter 7 Preview
“Solving and Graphing Linear Systems”
(7.1) Solve Linear Systems by Graphing
(7.2) Solve Linear Systems by Substituting
(7.3) Solve Linear Systems by Adding or Subtracting
(7.4) Solve Linear Systems by Multiplying First
(7.5) Solve Special Types of Linear Systems
(7.6) Solve Systems of Linear Inequalities
Section 7.1
“Solve Linear Systems by Graphing”
Linear System–
consists of two or more linear equations.
x + 2y = 7
3x – 2y = 5
Equation 1
Equation 2
A solution to a linear system is an ordered
pair (a point) where the two linear equations
(lines) intersect (cross).
Solving a Linear System by Graphing
(1) Graph both equations in the same plane.
(2) Estimate the coordinates of the point where the two
lines intersect.
(3) Check the coordinate by substituting into EACH
equation of the linear system, to see if the point is a
solution for both equations.
Using a Graph to Solve a Linear System
Use the graph to solve the
system. Then check your
solution algebraically.
x + 2y = 7
Equation 1
3x – 2y = 5
Equation 2
SOLUTION
The lines appear to intersect at the point (3, 2).
CHECK
Substitute 3 for x and 2 for y in each equation.
Equation 1
x + 2y = 7
?
3 + 2(2) = 7
7=7
Equation 2
3x – 2y = 5
?
3(3) – 2(2) = 5
5=5
Because the ordered pair
(3, 2) is a solution of
each equation, it is a
solution of the system.
Using a Graph to Solve a Linear System
Use the graph to solve the
system. Then check your
solution algebraically.
x + 4y = -8
Equation 1
-x + y = -7
Equation 2
SOLUTION
The lines appear to intersect at the point (4, -3).
CHECK
Substitute 4 for x and -3 for y in each equation.
Equation 1
x + 4y = -8
?
4 + 4(-3)= -8
-8 = -8
Equation 2
-x + y = -7
?
-4 + (-3) = -7
-7= -7
Because the ordered pair
(4, -3) is a solution of
each equation, it is a
solution of the system.
Standardized Test Practice
The parks and recreation department in your town
offers a season pass for $90. As a season pass holder, you
pay $4 per session to use the town’s tennis courts.
Without the season pass, you pay $13 per session to use
the tennis courts.
Which system of equations can be used to find the number x of sessions of
tennis after which the total cost y with a season pass, including the cost of the
pass, is the same as the total cost without a season pass?
A y = 4x
y = 13x
B y = 4x
y = 90 + 13x
C y = 13x
y = 90 + 4x
D y = 90 + 4x
y = 90 + 13x
Standardized Test Practice
Which system of equations can be used to find the number x of sessions of
tennis after which the total cost y with a season pass, including the cost of the
pass, is the same as the total cost without a season pass?
EQUATION 1
y
=
13
x
EQUATION 2
y
=
90
+
4
x
A y = 4x
y = 13x
B y = 4x
y = 90 + 13x
C y = 13x
y = 90 + 4x
D y = 90 + 4x
y = 90 + 13x
Solve a multi-step problem
A business rents in-line skates and
bicycles. During one day, the business
has a total of 25 rentals and collects
$450 for the rentals. Find the number of
pairs of skates rented and the number
of bicycles rented.
STEP 1
Write a linear system. Let x be the number of pairs
of skates rented, and let y be the number of
bicycles rented.
x + y =25
Equation for number of rentals
15x + 30y = 450 Equation for money collected from rentals
Solve a multi-step problem
STEP 2
Graph both equations.
STEP 3
Estimate the point of intersection. The
two lines appear to intersect at (20, 5).
STEP 4
Check whether (20, 5) is a solution.
20 + 5=? 25
25 = 25
15(20) + 30(5) =? 450
450 = 450
ANSWER
The business rented 20 pairs of skates and 5 bicycles.
Section 7.2
“Solve Linear Systems by Substitution”
Solving a Linear System by Substitution
(1) Solve one of the equations for one of its variables. (When possible,
solve for a variable that has a coefficient of 1 or -1).
(2) Substitute the expression from step 1 into the other equation and solve
for the other variable.
(3) Substitute the value from step 2 into the revised equation from step 1 and
solve.
“Solve Linear Systems by Substituting”
y = 3x + 2
Equation 1
x + 2y = 11
Equation 2
x + 2(3x
2y = 11
+ 2) = 11
x + 6x + 4 = 11
7x + 4 = 11
x =1
Equation 1
y = 3x + 2
y = 3(1) + 2
y=5
(5) = 3(1) + 2
5=5
Substitute
Substitute value for
x into the original
equation
The solution is the point (1,5).
Substitute (1,5) into both
equations to check.
(1) + 2(5) = 11
11 = 11
“Solve Linear Systems by Substituting”
Equation 1
x – 2y = -6
Equation 2
4x + 6y = 4
x = -6 + 2y
4(-6+ +6y2y)
4x
= 4+ 6y = 4
-24 + 8y + 6y = 4
-24 + 14y = 4
y =2
x – 2y = -6
Equation 1
x = -6 + 2(2)
x = -2
(-2) - 2(2) = -6
-6 = -6
Substitute
Substitute value for
x into the original
equation
The solution is the point (-2,2).
Substitute (-2,2) into both
equations to check.
4(-2) + 6(2) = 4
4=4
Solve a multi-step problem
A business rents in-line skates and
bicycles. During one day, the business
has a total of 25 rentals and collects
$450 for the rentals. Find the number of
pairs of skates rented and the number
of bicycles rented.
STEP 1
Write a linear system. Let x be the number of pairs
of skates rented, and let y be the number of
bicycles rented.
x + y =25
Equation for number of rentals
15x + 30y = 450 Equation for money collected from rentals
Solve a multi-step problem
STEP 2
Solve equation 1 for x.
Equation 1
x + y = 25
Equation 2
15x + 30y = 450
x = 25 - y
Substitute
15x + 30y
15(25
- y) +
= 30y
450 = 450
375 - 15y + 30y = 450
375 + 15y = 450
15y = 75
y =5
x + y = 25
ANSWER
Equation 1
x + (5) = 25
x = 20
Substitute value for
x into the original
equation
The business rented 20 pairs of skates and 5 bicycles.
During a football game, a bag of popcorn
sells for $2.50 and a pretzel sells for $2.00.
The total amount of money collected during the
game was $336. Twice as many bags of
popcorn sold compared to pretzels. How
many bags of popcorn and pretzels were sold
during the game?
y = 2x
x=
$2.50y + $2.00x = $336
y=
96 bags of popcorn and 48 pretzels
NJASK7 prep
Homework
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Text p. 430, #4-20, 30-34 evens
Text p. 439, #4-24 even, #32, #36