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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

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
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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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