To identify the correct graph of the system of equations given by:
[tex]\[
\begin{cases}
3x - y = 12 \\
x + 4y = 4
\end{cases}
\][/tex]
we need to find the point of intersection of the two lines represented by these equations. This intersection point will be the solution to the system.
### Step-by-Step Solution:
1. Solve the system of equations:
[tex]\[
3x - y = 12 \tag{1}
\][/tex]
[tex]\[
x + 4y = 4 \tag{2}
\][/tex]
We will solve these equations simultaneously.
2. From Equation (2), express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = 4 - 4y
\][/tex]
3. Substitute [tex]\( x = 4 - 4y \)[/tex] into Equation (1):
[tex]\[
3(4 - 4y) - y = 12
\][/tex]
Simplify:
[tex]\[
12 - 12y - y = 12
\][/tex]
[tex]\[
12 - 13y = 12
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
-13y = 0
\][/tex]
[tex]\[
y = 0
\][/tex]
4. Substitute [tex]\( y = 0 \)[/tex] back into [tex]\( x = 4 - 4y \)[/tex]:
[tex]\[
x = 4 - 4(0)
\][/tex]
[tex]\[
x = 4
\][/tex]
The solution to the system of equations is [tex]\( x = 4 \)[/tex] and [tex]\( y = 0 \)[/tex]. So, the point of intersection of the two lines is [tex]\( (4, 0) \)[/tex].
5. Graph both equations on the coordinate plane:
- For the first equation [tex]\( 3x - y = 12 \)[/tex]:
- If [tex]\( x = 0 \)[/tex]:
[tex]\[
-y = 12 \implies y = -12 \quad \text{(Point: (0, -12))}
\][/tex]
- If [tex]\( y = 0 \)[/tex]:
[tex]\[
3x = 12 \implies x = 4 \quad \text{(Point: (4, 0))}
\][/tex]
- For the second equation [tex]\( x + 4y = 4 \)[/tex]:
- If [tex]\( x = 0 \)[/tex]:
[tex]\[
4y = 4 \implies y = 1 \quad \text{(Point: (0, 1))}
\][/tex]
- If [tex]\( y = 0 \)[/tex]:
[tex]\[
x = 4 \quad \text{(Point: (4, 0))}
\][/tex]
When you plot these points and draw the lines, you will find that they intersect at the point [tex]\( (4, 0) \)[/tex], confirming that this is the correct solution.
### Summary:
- The point of intersection of the given system of equations [tex]\( 3x - y = 12 \)[/tex] and [tex]\( x + 4y = 4 \)[/tex] is [tex]\( (4, 0) \)[/tex].
- The correct graph will show the two lines intersecting at the point [tex]\( (4, 0) \)[/tex].
Make sure to identify the graph that shows this intersection point.
