To graphically represent and solve the given system of equations, we follow these steps:
1. Write down the equations:
[tex]\[
\left\{
\begin{array}{l}
x - 2y = 4 \\
y = 2x + 4
\end{array}
\right.
\][/tex]
2. Find the points of intersection:
These are the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations. Here, we find that the unique solution is [tex]\((x, y) = (-4, -4)\)[/tex].
3. Graph the first equation [tex]\(x - 2y = 4\)[/tex]:
To graph this, we express it in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
x - 2y = 4 \\
-2y = -x + 4 \\
y = \frac{1}{2}x - 2
\][/tex]
- Intercept Method:
- When [tex]\(x = 0\)[/tex]: [tex]\[
y = -2
\][/tex]
- When [tex]\(y = 0\)[/tex]: [tex]\[
0 = \frac{1}{2}x - 2 \implies x = 4
\][/tex]
Plot the points [tex]\((0, -2)\)[/tex] and [tex]\((4, 0)\)[/tex] on the graph, and draw a line through these points.
4. Graph the second equation [tex]\(y = 2x + 4\)[/tex]:
This is already in slope-intercept form [tex]\(y = mx + b\)[/tex], where the slope [tex]\(m = 2\)[/tex] and the y-intercept [tex]\(b = 4\)[/tex].
- Intercept Method:
- When [tex]\(x = 0\)[/tex]: [tex]\[
y = 4
\][/tex]
To find another point, select [tex]\(x = 1\)[/tex]:
- When [tex]\(x = 1\)[/tex]: [tex]\[
y = 2(1) + 4 = 6
\][/tex]
Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 6)\)[/tex] on the graph, and draw a line through these points.
5. Identify the intersection:
The two lines intersect at the point [tex]\((-4, -4)\)[/tex].
6. Conclusion:
The graph should clearly show the two lines [tex]\(y = \frac{1}{2}x - 2\)[/tex] and [tex]\(y = 2x + 4\)[/tex] intersecting at the point [tex]\((-4, -4)\)[/tex]. This means [tex]\((-4, -4)\)[/tex] is the solution to the system of equations.
Thus, the correct graph would display the lines intersecting at the point [tex]\((-4, -4)\)[/tex], demonstrating that [tex]\((x, y) = (-4, -4)\)[/tex] is the solution to the system of equations.
