Question

Choose the correct equation type for each system. ry = 3x - 2 |3x-y=4 Consistent S-y = -x +5 12y = -4x + 20 y = -x + 4 [x=-y-6 > y = 4x + 2 y = 6x - 10 (y=2x+1 y = -2x + 3 =+3 - 2x + 5y = 0 y = x

Answer 1

Let's solve each system to check if it is a consistent or inconsisent system of equations:

a.

`\begin{gathered} \begin{cases}y=3x-2 \\ 3x-y=4\end{cases} \\ \text{ Substituting the value of y from the first equation in the second:} \\ 3x-(3x-2)=4 \\ 3x-3x+2=4 \\ 2=4 \end{gathered}`

This system has no solution, therefore it is a inconsistent system.

b.

`\begin{gathered} \begin{cases}(1)/(2)y=-x+5 \\ 2y=-4x+20\end{cases} \\ \text{Multiplying the first equation by 4:} \\ 2y=-4x+20 \end{gathered}`

Both equations represent the same line, so the system has infinite solutions, therefore it is a consistent dependent system.

c.

`\begin{gathered} \begin{cases}y=-x+4 \\ x=-y-6\end{cases} \\ \text{ Applying the value of y from the first equation in the second one:} \\ x=-(-x+4)-6 \\ x=x-4-6 \\ 0=-10 \end{gathered}`

This system has no solution, therefore it is a inconsistent system.

d.

`\begin{gathered} \begin{cases}y=4x+2 \\ y=6x-10\end{cases} \\ \text{Comparing the y values:} \\ 4x+2=6x-10 \\ 6x-4x=2+10 \\ 2x=12 \\ x=6 \\ \\ y=4\cdot6+2=24+2=26 \end{gathered}`

This system has one solution, therefore it is a consistent independent system.

e.

`\begin{gathered} \begin{cases}y=2x+1 \\ y=-2x+3\end{cases} \\ \text{Comparing the y values:} \\ 2x+1=-2x+3 \\ 2x+2x=3-1 \\ 4x=2 \\ x=(1)/(2) \\ \\ y=2\cdot(1)/(2)+1=1+1=2 \end{gathered}`

This system has one solution, therefore it is a consistent independent system.

f.

`\begin{gathered} \begin{cases}-2x+5y=0 \\ y=(2)/(5)x\end{cases} \\ \text{ Applying the value of y from the second equation in the first one:} \\ -2x+5\cdot((2)/(5)x)=0 \\ -2x+2x=0 \\ 0=0 \end{gathered}`

This system has infinite solutions (the equations represent the same line), therefore it is a consistent dependent system.