Question

Which equation, added to the graphed line, completes a system of equations with infinitely many solutions?3x−2y=−16x−9y=−99x−6y=−62x−3y=1

Answer 1

The equation that would be added to the graph line that will make the system of equations has infinitely many solutions, will be the equation that has the same graph has the graph provided on the graph sheet.

Thus, to find the equation, we will need to plot all the equations and find the graph that corresponds to the graph on the graph sheet.

First equation

`\begin{gathered} 3x-2y=-1 \\ y-\text{intercept}\Rightarrow\text{set x=0} \\ 3(0)-2y=-1 \\ -2y=-1 \\ y=(-1)/(-2)=(1)/(2) \\ y-\text{intercept}\Rightarrow(0,(1)/(2)) \\ x-\text{intercept}\Rightarrow\text{ set y=0} \\ 3x-2(0)=-1_{} \\ 3x=-1 \\ x=(-1)/(3) \\ x-\text{intercept is }\Rightarrow(-(1)/(3),0) \end{gathered}`

The graph of the equation above is shown below

For the second equation

`\begin{gathered} 6x-9y=-9 \\ y-\text{intercept}\Rightarrow\text{set x=0} \\ 6(0)-9y=-9 \\ -9y=-9 \\ y=(-9)/(-9)=1 \\ y-\text{intercept}\Rightarrow(0,1) \\ x-\text{intercep}\Rightarrow\text{ set y=0} \\ 6x-9(0)=-9 \\ 6x=-9 \\ x=(-9)/(6)=-(3)/(2) \\ x-intercept\Rightarrow(-(3)/(2),0) \end{gathered}`

The graph of the second equation is

For the third equation

`\begin{gathered} 9x-6y=-6 \\ y-\text{intercept}\Rightarrow\text{set x=0} \\ 9(0)-6y=-6 \\ -6y=-6 \\ y=(-6)/(-6)=1 \\ y-\text{intercept}\Rightarrow(0,1) \\ x-\text{intercept}\Rightarrow\text{set y=0} \\ 9x-6(0)=-6 \\ 9x=-6 \\ x=(-6)/(9) \\ x=-(2)/(3) \\ x-\text{intercept}\Rightarrow(-(2)/(3),0) \end{gathered}`

The graph of the third equation is

For the fourth equation

`\begin{gathered} 2x-3y=1 \\ y-\text{intercept}\Rightarrow\text{set x=0} \\ 2(0)-3y=1 \\ -3y=1 \\ y=-(1)/(3) \\ y-\text{intercept}\Rightarrow(0,-(1)/(3)) \\ x-\text{intercept}\Rightarrow\text{set y=0} \\ 2x-3(0)=1 \\ 2x=1 \\ x=(1)/(2) \\ x-\text{intercept}\Rightarrow((1)/(2),0) \end{gathered}`

The graph of the fourth equation is

Hence, the graphs above, the equation that aligns with the graph provided on the graph sheet, that will make the system of equation has infinitely many solution is

`6x-9y=-9`