To solve the system of equations graphically and identify the solutions, we need to:
1. Understand the equations provided:
- The first equation \(x^2 + y^2 = 49\) represents a circle with radius 7 centered at the origin (0,0).
- The second equation \(x = -y - 7\) represents a line with a slope of -1 and a y-intercept at -7.
2. Graph both equations on the coordinate plane.
### Step-by-Step Solution:
#### Step 1: Graph the Circle
- The equation \(x^2 + y^2 = 49\) describes a circle centered at (0,0) with a radius of 7.
- Plot a circle with its center at (0, 0) and passing through (7, 0), (-7, 0), (0, 7), and (0, -7).
#### Step 2: Graph the Line
- The equation \(x = -y - 7\) can be rewritten as \(y = -x - 7\) for graphing purposes. The slope is -1 and the y-intercept is -7.
- When \(y = 0\), solve for \(x\):
[tex]\[
0 = -x - 7 \implies x = -7
\][/tex]
So, one point on the line is \((-7, 0)\).
- When \(x = 0\), solve for \(y\):
[tex]\[
x = -y - 7 \implies 0 = -y - 7 \implies y = -7
\][/tex]
So, another point on the line is \((0, -7)\).
- Plot the line passing through the points \((-7, 0)\) and \((0, -7)\).
#### Step 3: Identify Intersection Points
- The solutions to the system of equations are the points where the circle and the line intersect.
- From our detailed calculations (aligning with the correct solutions), we see that the circle \(x^2 + y^2 = 49\) and the line \(x = -y - 7\) intersect at the points \((-7, 0)\) and \((0, -7)\).
### Graph:
1. Plot the circle centered at (0,0) with a radius of 7.
2. Plot and draw the line passing through the points \((-7, 0)\) and \((0, -7)\).
3. Mark the points of intersection at \((-7, 0)\) and \((0, -7)\).
These points [tex]\((-7, 0)\)[/tex] and [tex]\((0, -7)\)[/tex] are the solutions to the system of equations.
