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Elementary Algebra
Systems of Linear Equations

1
Solving Linear Systems by Graphing

Problem  1

Determine whether each ordered pair is a solution to the following system:

\[\begin{aligned} x+y &= 3\\ 3x+4y &= 2\end{aligned}\]

  1. \((-2,5)\)

  2. \((10,-7)\)

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Stephanie cc
Stephanie
Nathan cc
Nathan
Breylor cc
Breylor
Octabio espanol spanish
Octabio
Problem  2
practice icon

Solve

\[\begin{aligned} x+y &= 4\\ x-y &= -2\end{aligned}\]

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Preston cc
Preston
Edwin espanol spanish
Edwin
Problem  3
practice icon

Solve

\[\begin{aligned} x+2y &= 8\\ 2x-3y &= 2\end{aligned}\]

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Arielle cc
Arielle
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  4

Solve the following system by graphing:

\[\begin{aligned} y &= 2x-3\\ x &= 3\end{aligned}\]

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Preston cc
Preston
Betsy cc
Betsy
Aaron cc
Aaron
Cynthia espanol spanish
Cynthia
Problem  5

Solve the following system by graphing:

\[\begin{aligned} y &= x-2\\ y &= x+1\end{aligned}\]

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Preston cc
Preston
Betsy cc
Betsy
Molly S. cc
Molly S.
Edwin espanol spanish
Edwin
Problem  6
practice icon

Graph the system.

\[\begin{aligned} 2x+y &= 4\\ 4x+2y &= 8\end{aligned}\]

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Preston cc
Preston
Betsy cc
Betsy
Molly S. cc
Molly S.
Edwin espanol spanish
Edwin
Mini Lecture

Mini Lecture
Solve each system.

  1. \(\begin{aligned}[t] x + y &= 3 \\ x-y &= 1 \end{aligned}\)

  2. \(\begin{aligned}[t] 3 x -2 y &= 6 \\ x-y &= 1 \end{aligned}\)

  3. \(\begin{aligned}[t] x + y &= 2 \\ x &= -3 \end{aligned}\)

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Mr. McKeague cc
Mr. McKeague

2
The Elimination Method

Problem  1
practice icon

Solve

\[\begin{aligned} x+y&=4\\ x-y&=2\end{aligned}\]

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Matt cc
Matt
Anthony espanol spanish
Anthony
Problem  2
practice icon

Solve

\[\begin{aligned} x+2y&=4\\ x-y&=-5\end{aligned}\]

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Mr. McKeague cc
Mr. McKeague
Brooke cc
Brooke
Aaron cc
Aaron
Anthony espanol spanish
Anthony
Study Skill

Common Mistake This is a mistake in using notation incorrectly.

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Mr. McKeague cc
Mr. McKeague
Problem  4
practice icon

Solve

\[\begin{aligned} 2x-y&=6\\ x+3y&=3\end{aligned}\]

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Mr. McKeague cc
Mr. McKeague
Aaron cc
Aaron
Anthony espanol spanish
Anthony
Problem  5

Solve the system.

\[\begin{aligned} 2x+4y &= -1\\ 3x+5y &= -2\end{aligned}\]

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Stephanie
Stephanie
Nathan cc
Nathan
Breylor cc
Breylor
Octabio espanol spanish
Octabio
Problem  6

Solve the system.

\[\begin{aligned} 3x+5y &= 2\\ 5x+4y &= 1\end{aligned}\]

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Nathan cc
Nathan
Breylor cc
Breylor
Octabio espanol spanish
Octabio
Problem  7
practice icon

Solve

\[\begin{aligned} \displaystyle\frac{1}{2}x-\displaystyle\frac{1}{3}y&=2\\ \displaystyle\frac{1}{4}x+\displaystyle\frac{2}{3}y&=6\end{aligned}\]

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Molly S. cc
Molly S.
Edwin espanol spanish
Edwin
Problem  8
practice icon

Solve

\[\begin{aligned} 2x-y&=2\\ 4x-2y&=12\end{aligned}\]

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Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  9
practice icon

Solve

\[\begin{aligned} 4x-3y&=2\\ 8x-6y&=4\end{aligned}\]

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Mini Lecture

Mini Lecture
Solve each system.

  1. \(\begin{aligned}[t] x + y &= 3 \\ x-y &= 1 \end{aligned}\)

  2. \(\begin{aligned}[t] 3 x - y &= 4 \\ 2x+2y &= 24 \end{aligned}\)

  3. \(\begin{aligned}[t] 2x + 9y &= 2 \\ 5x+3y &= -8 \end{aligned}\)

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Mr. McKeague cc
Mr. McKeague

3
The Substitution Method

Problem  1
practice icon

Solve
\(x+y=2\)
\(y=2x-1\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Aaron cc
Aaron
Cynthia espanol spanish
Cynthia
Problem  2

Solve the following system by the substitution method:

\[\begin{aligned} 2x-3y&= 12\\ x&= 2y+7\end{aligned}\]

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Stephanie cc
Stephanie
Nathan cc
Nathan
Breylor cc
Breylor
Octabio espanol spanish
Octabio
Problem  3
practice icon

Solve
\(x-3y=-1\)
\(2x-3y=4\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  4
practice icon

Solve
\(-2x+4y=14\)
\(-3x+y=6\)

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CJ cc
CJ
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  5
practice icon

Solve
\(4x+2y=8\)
\(y=-2x+4\)

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Mr. McKeague cc
Mr. McKeague
Betsy cc
Betsy
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Problem  6

There are two cell phone contract rates, at how many minutes will the two contracts cost the same?

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CJ cc
CJ
Stefanie cc
Stefanie
Aaron cc
Aaron
Edwin espanol spanish
Edwin
Mini Lecture

Mini Lecture
Solve each system.

  1. \(\begin{aligned}[t] x + y &= 11 \\ y &= 1x-1 \end{aligned}\)

  2. \(\begin{aligned}[t] 2 x + y &= 1 \\ x-5y &= 17 \end{aligned}\)

  3. \(\begin{aligned}[t] 7x - 6y &= -1 \\ x &= 2y-1 \end{aligned}\)

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Mr. McKeague cc
Mr. McKeague

4
Applications

Problem  1
practice icon

One number is \(2\) more than \(5\) times another number. Their sum is \(20\). Find the two numbers.

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Betsy cc
Betsy
Molly S. cc
Molly S.
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  2
practice icon

Mr. Hicks had \(\$15,000\) to invest. He invested part at \(6\%\) and the rest at \(7\%\). If he earns \(\$980\) in interest, how much did he invest in each account?

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Betsy cc
Betsy
Molly S. cc
Molly S.
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  3
practice icon

John has \(\$1.70\) all in dimes and nickels. He has a total of \(22\) coins. How many of each kind does he have?

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Betsy cc
Betsy
Molly S. cc
Molly S.
Aaron cc
Aaron
Gordon espanol spanish
Gordon
Problem  4
practice icon

How much \(20\%\) alcohol solution and \(50\%\) alcohol solution must be mixed to get \(12\) gallons of \(30\%\) alcohol solution?

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Betsy cc
Betsy
Aaron cc
Aaron
Cynthia espanol spanish
Cynthia
Study Skill

SAT Exam: Multiple Choice Question

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Mr. McKeague cc
Mr. McKeague
Mini Lecture

Mini Lecture

  1. Two positive numbers have a difference of \(5\). The larger is one more than twice the smaller. Find the two numbers.

  2. A woman invests \(4\) times as much money at \(5\%\) than she does at \(6\%\). The total interest for the year is \(\$520\). How much is invested at each rate?

  3. Tom has \(21\) coins totaling \(\$3.45\). The coins are dimes and quarters. How many of each type does he have?

  4. How many liters of \(50\%\) alcohol solution and \(20\%\) alcohol solution must be mixed to obtain \(18\) liters of \(30\%\) solution?

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Mr. McKeague cc
Mr. McKeague