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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

hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Kendra Reply
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
yeah
Asamoah
Quadratic equations involving factorization
Winner Reply
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
Please help me solve this 10x+14=-2x+38
Erica
the previous equation should be 3x = 1/6 x=1/18
Sriram
for the new one 10x + 2x = 38 - 14
Sriram
12x = 24 x=2
Sriram
10x + 14 = -2x +38 10x + 2x = 38 - 14 12x = 24 divide both sides by the coefficient of x, which is 12 therefore × = 2
vida
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
really
Asamoah
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Ok cool answer peggie
Frank
thanks
Ramil
x=7/2 y=6/7
Asamoah
Thanks , all of you are correct.
Joseph

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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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