Question

Based on the given graph, determine the number of solutions to the system of equations

Answer 1

`\begin{cases}y=(x-2)^2-3\\y=-(x-1)^2+2\end{cases}\qquad \left(\:\boxed{\sf 0,1}\:\right)\; \sf and \; \left(\:\boxed{\sf 3, -2}\:\right)`

`\begin{cases}y=-(x-1)^2+2\\y=(x-1)^2+2\end{cases} \qquad \left(\:\boxed{1,2}\:\right)\; \sf and \; \left(\:\boxed{\sf none}\:\right)`

`\begin{cases}y=(x+2)^2+5\\y=-(x-1)^2+2\end{cases}\qquad \left(\:\boxed{\sf none}\:\right)\; \sf and \; \left(\:\boxed{\sf none}\:\right)`

Explanation:

The points where the graphs of the equations intersect are the solutions to a graphed system of equations.

Therefore, to determine the number of solutions to the given graphed systems of equations, observe the number of points of intersection between the graphs.

`\hrulefill`

Given system of equations:

`\begin{cases}y=(x-2)^2-3\\y=-(x-1)^2+2\end{cases}`

From observation of the given graph of this system of equations, there are 2 points of intersection, so there are 2 solutions.

The 2 solutions are:

• (0, 1) and (3, -2)

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Given system of equations:

`\begin{cases}y=-(x-1)^2+2\\y=(x-1)^2+2\end{cases}`

From observation of the given graph of this system of equations, there is one point of intersection, so there is one solution.

The solution is:

• (2, 1)

`\hrulefill`

Given system of equations:

`\begin{cases}y=(x+2)^2+5\\y=-(x-1)^2+2\end{cases}`

From observation of the given graph of this system of equations, there are no points of intersection, so there are no solutions.