Question

Y = 2x - 9 y = -1/2x + 1Graph both equations to find the solutionfor this system.

Answer 1

To answer this question, we can graph both lines equations using the intercepts of both lines. The intercepts are the x- and the y-intercepts for both lines.

The x-intercept is the point where the line passes through the x-axis. At this point, y = 0. Likewise, the y-intercept is the point where the line passes through the y-axis. At this point, x = 0.

Therefore, we can proceed as follows:

1. Graphing the line y = 2x - 9

First, we can find the x-intercept. For this, y = 0.

`\begin{gathered} y=2x-9\Rightarrow y=0 \\ 0=2x-9 \\ 9=2x \\ (9)/(2)=(2)/(2)x \\ (9)/(2)=x\Rightarrow x=(9)/(2)=4.5 \end{gathered}`

Therefore, the x-intercept is (4.5, 0).

The y-intercept is:

`y=2(0)-9\Rightarrow y=-9`

Therefore, the y-intercept is (0, -9).

With these two points (4.5, 0) and (0, -9) we can graph the line y = 2x - 9.

2. Graphing the line y = -(1/2)x +1

We can proceed similarly here.

Finding the x-intercept:

`\begin{gathered} 0=-(1)/(2)x+1 \\ (1)/(2)x=1 \\ 2\cdot(1)/(2)x=2\cdot1 \\ (2)/(2)x=2\Rightarrow x=2 \end{gathered}`

Therefore, the x-intercept is (2, 0).

Finding the y-intercept:

`\begin{gathered} y=-(1)/(2)(0)+1 \\ y=1 \end{gathered}`

Then the y-intercept is (0, 1).

Now we can graph this line by using the points (2, 0) and (0, 1).

Graphing both lines

To graph the line y = 2x - 9, we have the following coordinates (4.5, 0) and (0, -9) ---> Red line.

To graph the line y = -(1/2)x + 1, we have the coordinates (2, 0) and (0, 1) ---> Blue line.

We graph both lines, and the point where the two lines intersect will be the solution of the system:

We can see that the point where the two lines intersect is the point (4, -1). Therefore, the solution for this system is (4, -1).

We can check this if we substitute the solution into the original equations as follows:

`\begin{gathered} y=2x-9 \\ -2x+y=-9\Rightarrow x=4,y=-1 \\ -2(4)+(-1)=-9 \\ -8-1=-9 \\ -9=-9\Rightarrow This\text{ is True.} \end{gathered}`

And

`\begin{gathered} y=-(1)/(2)x+1 \\ (1)/(2)x+y=1\Rightarrow x=4,y=-1 \\ (1)/(2)(4)+(-1)=1 \\ 2-1=1 \\ 1=1\Rightarrow This\text{ is True.} \end{gathered}`

In summary, we found the solution of the system:

`\begin{gathered} \begin{cases}y=2x-9 \\ y=-(1)/(2)x+1\end{cases} \\ \end{gathered}`

Using the intercepts of the lines, graphing the lines, and the point where the two lines intersect is the solution for the system. In this case, the solution is (4, -1) or x = 4, and y = -1.