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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

In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
Jenise Reply
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
jamie Reply
6t+3
Melissa
6t +3
Bollywood
Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary?
Iris Reply
let us suppose her original salary is 'm'. so, according to the given condition, m*(6/100)=30 m= (30*100)/6 m= 500 hence, her original salary is $500.
Simply
28.50
Toi
thanks
Jeorge
How many pounds of nuts selling for $6 per pound and raisins selling for $3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound?
Valeria Reply
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that code $20 per square foot. How many square feet of each tile should she use so that the overal cost of he backsplash will be $10 per square foot?
Nia Reply
I need help with maths can someone help me plz.. is there a wats app group?
Cindy Reply
WY need
Fernando
How did you get $750?
Laura Reply
if y= 2x+sinx what is dy÷dx
formon25 Reply
does it teach you how to do algebra if you don't know how
Kate Reply
Liam borrowed a total of $35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total $35,000? (Round your answer to the nearest tenth of a percent.)
Lara Reply
exact definition of length by bilbao
francemichael Reply
the definition of length
francemichael Reply
literal meaning of length
francemichael
exact meaning of length
francemichael
exact meaning of length
francemichael
how many typos can we find...?
Chris Reply
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
Chris Reply
4x+7y=29,x+3y=11 substitute method of linear equation
Srinu Reply
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce

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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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