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{ 2 x + 9 y = −4 3 x + 13 y = −7

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{ 1 3 x y = −3 x + 5 2 y = 2

( −3 , 2 )

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{ x + 1 2 y = 3 2 1 5 x 1 5 y = 3

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{ x + 1 3 y = −1 1 2 x 1 3 y = −2

( −2 , 3 )

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{ 1 3 x y = −3 2 3 x + 5 2 y = 3

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{ 2 x + y = 3 6 x + 3 y = 9

infinitely many solutions

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{ x 4 y = −1 −3 x + 12 y = 3

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{ −3 x y = 8 6 x + 2 y = −16

infinitely many solutions

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{ 4 x + 3 y = 2 20 x + 15 y = 10

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{ 3 x + 2 y = 6 −6 x 4 y = −12

infinitely many solutions

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{ 5 x 8 y = 12 10 x 16 y = 20

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{ −11 x + 12 y = 60 −22 x + 24 y = 90

inconsistent, no solution

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{ 7 x 9 y = 16 −21 x + 27 y = −24

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{ 5 x 3 y = 15 y = 5 3 x 2

inconsistent, no solution

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{ 2 x + 4 y = 7 y = 1 2 x 4

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Solve Applications of Systems of Equations by Elimination

In the following exercises, translate to a system of equations and solve.

The sum of two numbers is 65. Their difference is 25. Find the numbers.

The numbers are 20 and 45.

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The sum of two numbers is 37. Their difference is 9. Find the numbers.

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The sum of two numbers is −27. Their difference is −59. Find the numbers.

The numbers are 16 and −43.

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The sum of two numbers is −45. Their difference is −89. Find the numbers.

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Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for $114 or she is able to buy 2 shirts and 4 sweaters for $164. How much does a shirt cost? How much does a sweater cost?

A shirt costs $16 and a sweater costs $33.

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Peter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for $40 or he is able to buy 5 packages of paper and 6 staplers for $62. How much does a package of paper cost? How much does a stapler cost?

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The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?

There are 860 mg in a hot dog. There are 1,000 mg in a cup of cottage cheese.

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The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of calories in 5 hot dogs and 2 cups of cottage cheese is 1190 calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?

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Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.


{ 8 x 15 y = −32 6 x + 3 y = −5
{ x = 4 y 3 4 x 2 y = −6

elimination substitution

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{ y = 7 x 5 3 x 2 y = 16
{ 12 x 5 y = −42 3 x + 7 y = −15

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{ y = 4 x + 9 5 x 2 y = −21
{ 9 x 4 y = 24 3 x + 5 y = −14

substitution elimination

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{ 14 x 15 y = −30 7 x + 2 y = 10
{ x = 9 y 11 2 x 7 y = −27

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Everyday math

Norris can row 3 miles upstream against the current in the same amount of time it takes him to row 5 miles downstream, with the current. Solve the system. { r c = 3 r + c = 5

  1. for r , his rowing speed in still water.
  2. Then solve for c , the speed of the river current.

r = 4 c = 1

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Josie wants to make 10 pounds of trail mix using nuts and raisins, and she wants the total cost of the trail mix to be $54. Nuts cost $6 per pound and raisins cost $3 per pound. Solve the system { n + r = 10 6 n + 3 r = 54 to find n , the number of pounds of nuts, and r , the number of pounds of raisins she should use.

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Writing exercises

Solve the system
{ x + y = 10 5 x + 8 y = 56

by substitution
by graphing
Which method do you prefer? Why?

  1. (8, 2)

  2. This image is a graph that shows the solution to the system “x plus y equals 10” and 5x plus 8y equals 56. The solution is on an x, y coordinate plane. Two arrows intersect at points 8 and 2.
  3. Answers will vary.
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Solve the system
{ x + y = −12 y = 4 1 2 x

by substitution
by graphing
Which method do you prefer? Why?

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This figure shows a table with four rows and four columns. The columns are labeled, “I can…,” “Confidently.” “With some help.” and “No - I don’t get it.” The only column with filled in cells below it is labeled “I can…” It reads, “solve a system of equations by elimination.” “solve applications of systems of equations by elimination.” and “choose the most convenient method to solve a system of linear equations.”

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Questions & Answers

3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
don't understand answer
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
don't understand answer
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
32
Cindy
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
I am 63 and never learned algebra
Cindy
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
92
Cindy
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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