Question

consider the system of equations.\

$$\begin{cases}x^{2}+y^{2}=49\\\\x=-y - 7\\end{cases}$$

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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.

Explanation:

Step1: Substitute \( x = -y - 7 \) into \( x^2 + y^2 = 49 \)

Substitute \( x \) in the circle equation: \( (-y - 7)^2 + y^2 = 49 \)
Expand \( (-y - 7)^2 \): \( y^2 + 14y + 49 + y^2 = 49 \)
Combine like terms: \( 2y^2 + 14y + 49 - 49 = 0 \) → \( 2y^2 + 14y = 0 \)
Factor: \( 2y(y + 7) = 0 \)

Step2: Solve for \( y \)

Set each factor to zero:

• \( 2y = 0 \) → \( y = 0 \)
• \( y + 7 = 0 \) → \( y = -7 \)

Step3: Find corresponding \( x \) values

For \( y = 0 \): \( x = -0 - 7 = -7 \)
For \( y = -7 \): \( x = -(-7) - 7 = 7 - 7 = 0 \)

Check solutions in \( x^2 + y^2 = 49 \):

• For \( (-7, 0) \): \( (-7)^2 + 0^2 = 49 \), valid.
• For \( (0, -7) \): \( 0^2 + (-7)^2 = 49 \), valid.

Answer:

The solutions are the points \((-7, 0)\) and \((0, -7)\). To graph the line \( x = -y - 7 \) (or \( y = -x - 7 \)): find two points (e.g., when \( x = 0 \), \( y = -7 \); when \( y = 0 \), \( x = -7 \)), draw the line, then mark \((-7, 0)\) and \((0, -7)\) as intersection points.