To determine the solution to the system of equations [tex]\(4x + 4y = -8\)[/tex] and [tex]\(2x + 2y = -4\)[/tex], we'll follow these steps:
1. Simplify the Equations:
- First equation: [tex]\(4x + 4y = -8\)[/tex]
- Second equation: [tex]\(2x + 2y = -4\)[/tex]
Step-by-Step Simplification:
Divide the first equation by 4:
[tex]\[
\frac{4x + 4y}{4} = \frac{-8}{4}
\][/tex]
which simplifies to:
[tex]\[
x + y = -2
\][/tex]
Now divide the second equation by 2:
[tex]\[
\frac{2x + 2y}{2} = \frac{-4}{2}
\][/tex]
which simplifies to:
[tex]\[
x + y = -2
\][/tex]
2. Conclusion:
We see that both simplified equations are identical, [tex]\(x + y = -2\)[/tex]. This tells us that the two original equations represent the same line.
3. Graphing:
To graph [tex]\(x + y = -2\)[/tex]:
- Find the y-intercept when [tex]\(x = 0\)[/tex]:
[tex]\[
0 + y = -2 \implies y = -2 \quad \text{(Point: } (0, -2) \text{)}
\][/tex]
- Find the x-intercept when [tex]\(y = 0\)[/tex]:
[tex]\[
x + 0 = -2 \implies x = -2 \quad \text{(Point: } (-2, 0) \text{)}
\][/tex]
Plot these points on the graph and draw the line through them. Both [tex]\(4x + 4y = -8\)[/tex] and [tex]\(2x + 2y = -4\)[/tex] will graph to this same line.
4. Identify the Solution:
Since both equations describe the same line, they intersect at every point on that line. Therefore, they have infinitely many solutions.
Answer:
[tex]\[
\text{Infinitely Many Solutions}
\][/tex]
