Get personalized and accurate responses to your questions with IDNLearn.com. Join our interactive community and get comprehensive, reliable answers to all your questions.

Use the drawing tool(s) to form the correct answer on the provided graph.

Consider the system of equations:

[tex]\[
\begin{cases}
x^2 + y^2 = 49 \\
x = -y - 7
\end{cases}
\][/tex]

The first equation in the system is graphed below. Graph the linear equation on the coordinate plane and use the Mark Feature tool to place a point at the solution(s) of the system.


Sagot :

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.