To answer this question, we need to rewrite both equations in the slope-intercept form of the line. To achieve this, we can proceed as follows:
1. We need to isolate the variable y in both equations. In the first equation, we need to subtract 3x from both sides of the equation:
[tex]3x+2y=2\Rightarrow3x-3x+2y=2-3x\Rightarrow2y=2-3x[/tex]
And now, we need to divide both sides by 2 (to isolate the y variable):
[tex]\frac{2y}{2}=\frac{2}{2}-\frac{3x}{2}\Rightarrow y=1-\frac{3}{2}x\Rightarrow y=-\frac{3}{2}x+1[/tex]
2. We can proceed in a similar way to find the slope-intercept form of the other line:
[tex]2x-4y=12\Rightarrow2x-2x-4y=12-2x\Rightarrow-4y=12-2x[/tex]
Then, we have:
[tex]-\frac{4y}{4}=\frac{12}{4}-\frac{2}{4}x\Rightarrow-y=3-\frac{1}{2}x\Rightarrow y=\frac{1}{2}x-3[/tex]
Then, we have both line equations in the slope-intercept form.
Now, to graph both equations, we can use the coordinates of the x- and y-intercepts of both lines:
3. The x-intercept is the point when y = 0. Likewise, the y-intercept is the point when x = 0. We need to evaluate the equation in both cases. Then, we have:
First Line
For y = 0:
[tex]0=-\frac{3}{2}x+1\Rightarrow-1=-\frac{3}{2}x\Rightarrow1=\frac{3}{2}x\Rightarrow\frac{2}{3}\cdot1=\frac{2}{3}\cdot\frac{3}{2}x\Rightarrow\frac{2}{3}=x[/tex]
Then, we have the first coordinate: (2/3, 0).
For x = 0:
[tex]y=-\frac{3}{2}x+1\Rightarrow y=-\frac{3}{2}(0)+1\Rightarrow y=1[/tex]
Then, the other coordinate for this point is (0, 1)
To graph this line we can use these two points (0, 1) and (2/3, 0).
Second Line
We can proceed in a similar way to find these two coordinates for the second line:
For y = 0:
[tex]y=\frac{1}{2}x-3\Rightarrow0=\frac{1}{2}x-3\Rightarrow3=\frac{1}{2}x\Rightarrow2.3=2\cdot\frac{1}{2}x\Rightarrow6=x\Rightarrow x=6[/tex]
The x-intercept is (6, 0).
For x = 0:
[tex]y=\frac{1}{2}x-3\Rightarrow y=\frac{1}{2}\cdot(0)-3\Rightarrow y=-3[/tex]
The y-intercept is (0, -3).
To graph this line we can use these two points (0, -3) and (6, 0).
Therefore, the solution for this system of equations is x = 2, y = -2.
