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Exercises 6.5 Practice Problems

1.

Determine if each of the following is a solution to the equation \(4x+3y=12\text{:}\)

  1. \(\displaystyle (4,-1)\)

  2. \(\displaystyle (0,4)\)

  3. \(\displaystyle 3\)

Answer.

  1. \((4,-1)\) is not a solution

  2. \((0,4)\) is a solution

  3. \(3\) is not a solution

2.

Determine if each of the following is a solution to the equation \(x-5y=-2\text{:}\)

  1. \(\displaystyle (3,1)\)

  2. \(\displaystyle (0,0)\)

  3. \(\displaystyle \left(\frac{5}{2}, \frac{1}{2}\right)\)

Answer.

  1. \((3,1)\) is a solution

  2. \((0,0)\) is not a solution

  3. \(\left(\frac{5}{2}, \frac{1}{2}\right)\) is not a solution

3.

Suppose we have the system of equations:

\begin{equation*} \begin{cases} 4 \amp= 5x+y\\ -6 \amp= -2x+4y \end{cases} \end{equation*}

Check if each of the following is a solution to the system:

  1. \(\displaystyle (1,-1)\)

  2. \(\displaystyle (2,5)\)

  3. \(\displaystyle (-1,9)\)

Answer.

  1. \((1,-1)\) is a solution

  2. \((2,5)\) is not a solution

  3. \((-1,9)\) is not a solution

4.

Suppose we have the system of equations:

\begin{equation*} \begin{cases} 2 \amp= 3x-2y\\ 4 \amp= -2x+4y \end{cases} \end{equation*}

Check if each of the following is a solution to the system:

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

  2. \(\displaystyle (2,2)\)

  3. \(\displaystyle (0,0)\)

Answer.

  1. \((1,-2)\) is not a solution

  2. \((2,2)\) is a solution

  3. \((0,0)\) is not a solution

5.

Suppose we have the system of equations:

\begin{equation*} \begin{cases} 0 \amp= \frac{1}{2}x-2y\\ 12 \amp= 4x-4y \end{cases} \end{equation*}

Check if each of the following is a solution to the system:

  1. \(\displaystyle (1,-4)\)

  2. \(\displaystyle (2,3)\)

  3. \(\displaystyle (4,1)\)

Answer.

  1. \((1,-4)\) is not a solution

  2. \((2,3)\) is not a solution

  3. \((4,1)\) is a solution

6.

Suppose we have the system of equations:

\begin{equation*} \begin{cases} -3 \amp= 3x-2y\\ 0 \amp= 6x-2y \end{cases} \end{equation*}

Check if each of the following is a solution to the system:

  1. \(\displaystyle (1,3)\)

  2. \(\displaystyle (0,3)\)

  3. \(\displaystyle (-2,2)\)

Answer.

  1. \((1,3)\) is a solution

  2. \((0,3)\) is not a solution

  3. \((-2,2)\) is not a solution

7.

Solve the following system of equations using substitution:

\begin{equation*} \begin{cases}4 \amp= 4x-2y \\ 3 \amp= 5x-3y\end{cases} \end{equation*}
Answer.

\((3,4)\)

8.

Solve the following system of equations using substitution:

\begin{equation*} \begin{cases}-4 \amp= 2x-y \\ -2 \amp= 4x-3y\end{cases} \end{equation*}
Answer.

\((-5,-6)\)

9.

Solve the following system of equations using substitution:

\begin{equation*} \begin{cases}0 \amp= x-\frac{1}{2}y \\ -1 \amp= 3x-3y\end{cases} \end{equation*}
Answer.

\((\frac{1}{3},\frac{2}{3})\)

10.

Solve the following system of equations using substitution:

\begin{equation*} \begin{cases}10 \amp= 5x-2y \\ 2 \amp= x-4y\end{cases} \end{equation*}
Answer.

\((2,0)\)

11.

Solve the following system of equations using substitution:

\begin{equation*} \begin{cases}1 \amp= -3x+2y \\ 5 \amp= x-y\end{cases} \end{equation*}
Answer.

\((-11,-16)\)

12.

Solve the following system of equations using substitution:

\begin{equation*} \begin{cases}2 \amp= 4x-2y \\ 5 \amp= 2x+3y\end{cases} \end{equation*}
Answer.

\((1,1)\)

13.

Solve the following system of equations using substitution:

\begin{equation*} \begin{cases}-2 \amp= 4x-2y \\ 5 \amp= 2x+3y\end{cases} \end{equation*}
Answer.

\((\frac{1}{4},\frac{3}{2})\)

14.

Solve the following system of equations using elimination:

\begin{equation*} \begin{cases}4 \amp= 4x-2y \\ 3 \amp= 5x-3y\end{cases} \end{equation*}
Answer.

\((3,4)\)

15.

Solve the following system of equations using elimination:

\begin{equation*} \begin{cases}-4 \amp= 2x-y \\ -2 \amp= 4x-3y\end{cases} \end{equation*}
Answer.

\((-5,-6)\)

16.

Solve the following system of equations using elimination:

\begin{equation*} \begin{cases}0 \amp= x-\frac{1}{2}y \\ -1 \amp= 3x-3y\end{cases} \end{equation*}
Answer.

\((\frac{1}{3},\frac{2}{3})\)

17.

Solve the following system of equations using elimination:

\begin{equation*} \begin{cases}10 \amp= 5x-2y \\ 2 \amp= x-4y\end{cases} \end{equation*}
Answer.

\((2,0)\)

18.

Solve the following system of equations using elimination:

\begin{equation*} \begin{cases}1 \amp= -3x+2y \\ 5 \amp= x-y\end{cases} \end{equation*}
Answer.

\((-11,-16)\)

19.

Solve the following system of equations using elimination:

\begin{equation*} \begin{cases}2 \amp= 4x-2y \\ 5 \amp= 2x+3y\end{cases} \end{equation*}
Answer.

\((1,1)\)

20.

Solve the following system of equations using elimination:

\begin{equation*} \begin{cases}-2 \amp= 4x-2y \\ 5 \amp= 2x+3y\end{cases} \end{equation*}
Answer.

\((\frac{1}{4},\frac{3}{2})\)

21.

Solve the following system of equations:

\begin{equation*} \begin{cases}-2 \amp= 4x-2y \\ -1 \amp= 2x-y\end{cases} \end{equation*}
Answer.

Infinitely many solutions

22.

Solve the following system of equations:

\begin{equation*} \begin{cases}5 \amp= 3x-2y \\ -1 \amp= 4y-6x\end{cases} \end{equation*}
Answer.

No solution

23.

Solve the following system of equations:

\begin{equation*} \begin{cases}10 \amp= x-\frac{1}{2}y \\ -24 \amp= 2y-4x\end{cases} \end{equation*}
Answer.

No solution

24.

Solve the following system of equations:

\begin{equation*} \begin{cases}6 \amp= 2x-y \\ -12 \amp= 4x-2y\end{cases} \end{equation*}
Answer.

No solution

25.

Solve the following system of equations:

\begin{equation*} \begin{cases}6 \amp= 2x-y \\ 12 \amp= 4x-2y\end{cases} \end{equation*}
Answer.

Infinitely many solutions

26.

Solve the following system of equations:

\begin{equation*} \begin{cases}16 \amp= 4x-y \\ 8 \amp= 2x-\frac{1}{2}y\end{cases} \end{equation*}
Answer.

Infinitely many solutions

27.

Solve the following system of equations:

\begin{equation*} \begin{cases}21 \amp= -3x+7y \\ 7 \amp= -6x+14y\end{cases} \end{equation*}
Answer.

No solution