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Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manny need?

Manny needs 3 quarts juice concentrate and 9 quarts water.

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Alisha is making an 18 ounce coffee beverage that is made from brewed coffee and milk. The number of ounces of brewed coffee is 5 times greater than the number of ounces of milk. How many ounces of coffee and how many ounces of milk does Alisha need?

Alisha needs 15 ounces of coffee and 3 ounces of milk.

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Access these online resources for additional instruction and practice with solving systems of equations by graphing.

Key concepts

  • To solve a system of linear equations by graphing
    1. Graph the first equation.
    2. Graph the second equation on the same rectangular coordinate system.
    3. Determine whether the lines intersect, are parallel, or are the same line.
    4. Identify the solution to the system.
      If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.
      If the lines are parallel, the system has no solution.
      If the lines are the same, the system has an infinite number of solutions.
    5. Check the solution in both equations.

  • Determine the number of solutions from the graph of a linear system
    This table has two columns and four rows. The first row labels each column “Graph” and “Number of solutions.” Under “Graph” are “2 intersecting lines,” “Parallel lines,” and “Same line.” Under “Number of solutions” are “1,” “None,” and “Infinitely many.”
  • Determine the number of solutions of a linear system by looking at the slopes and intercepts
    This table is entitled “Number of Solutions of a Linear System of Equations.” There are four columns. The columns are labeled, “Slopes,” “Intercepts,” “Type of Lines,” “Number of Solutions.” Under “Slopes” are “Different,” “Same,” and “Same.” Under “Intercepts,” the first cell is blank, then the words “Different” and “Same” appear. Under “Types of Lines” are the words, “Intersecting,” “Parallel,” and “Coincident.” Under “Number of Solutions” are “1 point,” “No Solution,” and “Infinitely many solutions.”
  • Determine the number of solutions and how to classify a system of equations
    This table has four columns and four rows. The columns are labeled, “Lines,” “Intersecting,” “Parallel,” and “Coincident.” In the first row under the labeled column “lines” it reads “Number of solutions.” Reading across, it tell us that an intersecting line contains 1 point, a parallel line provides no solution, and a coincident line has infinitely many solutions. A consistent/inconsistent line has consistent lines if they are intersecting, inconsistent lines if they are parallel and consistent if the lines are coincident. Finally, dependent and independent lines are considered independent if the lines intersect, they are also independent if the lines are parallel, and they are dependent if the lines are coincident.

  • Problem Solving Strategy for Systems of Linear Equations
    1. Read the problem. Make sure all the words and ideas are understood.
    2. Identify what we are looking for.
    3. Name what we are looking for. Choose variables to represent those quantities.
    4. Translate into a system of equations.
    5. Solve the system of equations using good algebra techniques.
    6. Check the answer in the problem and make sure it makes sense.
    7. Answer the question with a complete sentence.

Practice makes perfect

Determine Whether an Ordered Pair is a Solution of a System of Equations . In the following exercises, determine if the following points are solutions to the given system of equations.

{ 2 x 6 y = 0 3 x 4 y = 5

( 3 , 1 ) ( −3 , 4 )

yes no

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{ 7 x 4 y = −1 −3 x 2 y = 1

( 1 , −2 )

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

( 4 , −3 ) ( 2 , 0 )

yes no

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{ −3 x + y = 8 x + 2 y = −9

( −5 , −7 ) ( −5 , 7 )

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{ x + y = 2 y = 3 4 x

( 8 7 , 6 7 ) ( 1 , 3 4 )

yes no

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{ x + y = 1 y = 2 5 x

( 5 7 , 2 7 ) ( 5 , 2 )

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{ x + 5 y = 10 y = 3 5 x + 1

( −10 , 4 ) ( 5 4 , 7 4 )

no yes

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

( −6 , 5 ) ( 5 , 4 3 )

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Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

{ 3 x + y = −3 2 x + 3 y = 5

( −2 , 3 )

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{ x + y = 2 2 x + y = −4

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{ −3 x + y = −1 2 x + y = 4

( 1 , 2 )

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{ −2 x + 3 y = −3 x + y = 4

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{ y = x + 2 y = −2 x + 2

( 0 , 2 )

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{ y = x 2 y = −3 x + 2

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

( 2 , 4 )

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

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{ x + y = −3 4 x + 4 y = 4

( 2 , −1 )

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{ x y = 3 2 x y = 4

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{ −3 x + y = −1 2 x + y = 4

( 1 , 2 )

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{ −3 x + y = −2 4 x 2 y = 6

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{ x + y = 5 2 x y = 4

( 3 , 2 )

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{ x y = 2 2 x y = 6

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{ x + y = 2 x y = 0

( 1 , 1 )

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{ x + y = 6 x y = −8

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{ x + y = −5 x y = 3

( −1 , −4 )

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{ x + y = −4 x + 2 y = −2

( 3 , 3 )

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{ x + 3 y = 3 x + 3 y = 3

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{ −2 x + 3 y = 3 x + 3 y = 12

( −5 , 6 )

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{ 2 x y = 4 2 x + 3 y = 12

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{ 2 x + 3 y = 6 y = −2

( 6 , −2 )

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Practice Key Terms 7

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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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