Transcript Algebra I. Chapter 7.1. Systems of Equations

Objective:
• To identify and write linear systems of
equations.
Writing a Linear Equation
A shipping clerk must send packages of
chemicals to a laboratory. The container
she is using to ship the chemicals will
hold 18 lbs. Each package of chemicals
weighs 3 lbs. How many packages of
chemicals will the container hold?
Systems of Equations
“Systems of equations” just means that we
are dealing with more than one EQUATION
and VARIABLE.
So far, we’ve basically just played around
with the equation for a line, which is y=mx
+ b.
Black Friday Shopping
You want to buy some chocolate candy for your math
teacher (ehem). The first website you find
(chocolateisamazing.com) charges $3 plus $1 per pound to
ship a box. The second website
(hersheycandyisthebest.com) charges $1 plus $2 per pound
to ship the same item.
What equations can you write to represent the situation?
Answer: For an object that weighs x pounds, the charges for
the two websites are represented by the equations y = x + 3
and y = 2x + 1.
System of Equation:
You are going to the mall with your friends and you have $200 to
spend from your recent birthday (or Christmas) money. You
discover a store that has all jeans for $25 and all dresses
for $50. You really, really want to take home 6 items of clothing
because you “need” that many new things.
Wouldn’t it be clever to find out how many pairs of jeans and how
many dresses you can buy so you use the whole $200 (tax not
included – your parents promised to pay the tax)?
System of Equation:
So what we want to know is how many pairs of jeans we want
to buy (let’s say “j”) and how many dresses we want to buy
(let’s say “d”). So always write down what your variables will
be:
•Let j = the number of
jeans you will buy
•Let d = the number
of dresses you’ll buy
System of Equation:
System of Equations:
• Two or more equations with the same set of
variables are called a system of equations.
• A solution of a system of equations is an
ordered pair that satisfies each equation in
the system.
EXAMPLE 1
Check the intersection point
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.
x + 2y = 7
?
3 + 2(2) = 7
7=7
EXAMPLE 1
Check the intersection point
3x – 2y = 5
?
3(3) – 2(2) = 5
5=5
ANSWER
Because the ordered pair (3, 2) is a solution of each
equation, it is a solution of the system.
EXAMPLE 2
Use the graph-and-check method
Solve the linear system:
–x + y = –7
Equation 1
x + 4y = –8
Equation 2
SOLUTION
STEP 1
Graph both equations.
EXAMPLE 2
Use the graph-and-check method
STEP 2
Estimate the point of intersection. The two lines
appear to intersect at (4, – 3).
STEP 3
Check whether (4, –3) is a solution by substituting 4 for
x and –3 for y in each of the original equations.
Equation 1
–x + y = –7
?
–(4) + (–3) = –7
–7 = –7
Equation 2
x + 4y = –8
?
4 + 4(–3) = –8
–8 = –8
EXAMPLE 2
Use the graph-and-check method
ANSWER
Because (4, –3) is a solution of each equation, it is a
solution of the linear system.
Use the graph-and-check
method
EXAMPLE
2
for Examples 1 and
2
GUIDED PRACTICE
Solve the linear system by graphing. Check your solution.
1. –5x + y = 0
5x + y = 10
ANSWER
(1, 5)
Use the graph-and-check
method
EXAMPLE
2
for Examples 1 and
2
GUIDED PRACTICE
Solve the linear system by graphing. Check your solution.
2. –x + 2y = 3
2x + y = 4
ANSWER
(1, 2)
Use the graph-and-check
method
EXAMPLE
2
for Examples 1 and
2
GUIDED PRACTICE
Solve the linear system by graphing. Check your solution.
3. x – y = 5
3x + y = 3
ANSWER
(2, 3)
EXAMPLE 3
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.
GUIDED PRACTICE
for Example 3
4. Solve the linear system in Example 3 to find the
number of sessions after which the total cost with a
season pass, including the cost of the pass, is the
same as the total cost without a season pass.
ANSWER
10 sessions
EXAMPLE 4
Solve a multi-step problem
RENTAL BUSINESS
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.
EXAMPLE 4
Solve a multi-step problem
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.