<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

{ 7 x 4 y = −1 −3 x 2 y = 1

( 1 , −2 )

Got questions? Get instant answers now!

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

( 4 , −3 ) ( 2 , 0 )

yes no

Got questions? Get instant answers now!

{ −3 x + y = 8 x + 2 y = −9

( −5 , −7 ) ( −5 , 7 )

Got questions? Get instant answers now!

{ x + y = 2 y = 3 4 x

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

yes no

Got questions? Get instant answers now!

{ x + y = 1 y = 2 5 x

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

Got questions? Get instant answers now!

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

( −10 , 4 ) ( 5 4 , 7 4 )

no yes

Got questions? Get instant answers now!

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

( −6 , 5 ) ( 5 , 4 3 )

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

{ x + y = 2 2 x + y = −4

Got questions? Get instant answers now!

{ −3 x + y = −1 2 x + y = 4

( 1 , 2 )

Got questions? Get instant answers now!

{ −2 x + 3 y = −3 x + y = 4

Got questions? Get instant answers now!

{ y = x + 2 y = −2 x + 2

( 0 , 2 )

Got questions? Get instant answers now!

{ y = x 2 y = −3 x + 2

Got questions? Get instant answers now!

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

( 2 , 4 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

{ x + y = −3 4 x + 4 y = 4

( 2 , −1 )

Got questions? Get instant answers now!

{ x y = 3 2 x y = 4

Got questions? Get instant answers now!

{ −3 x + y = −1 2 x + y = 4

( 1 , 2 )

Got questions? Get instant answers now!

{ −3 x + y = −2 4 x 2 y = 6

Got questions? Get instant answers now!

{ x + y = 5 2 x y = 4

( 3 , 2 )

Got questions? Get instant answers now!

{ x y = 2 2 x y = 6

Got questions? Get instant answers now!

{ x + y = 2 x y = 0

( 1 , 1 )

Got questions? Get instant answers now!

{ x + y = 6 x y = −8

Got questions? Get instant answers now!

{ x + y = −5 x y = 3

( −1 , −4 )

Got questions? Get instant answers now!

{ x + y = −4 x + 2 y = −2

( 3 , 3 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

{ −2 x + 3 y = 3 x + 3 y = 12

( −5 , 6 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

{ 2 x + 3 y = 6 y = −2

( 6 , −2 )

Got questions? Get instant answers now!
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask