Answered

The first three steps in determining the solution set of the system of equations, $y=-x^2-2 x+8$ and $y=2 x+11$, algebraically are shown in the table.




Step		Equation

Step 1		$-x^2-2 x+8=2 x+11$

Step 2		$0= x ^2+4 x +3$

Step 3		$0=( x +1)( x +3)$






Which represents the solution(s) of this system of equations?


A(-3, -1)
B(-3, 5)
C(-3, 5) and (-1, 9)
D$(1,13)$ and $(3,20)$

Verified Answer :

  • From the factored form \(0 = (x + 1)(x + 3)\), determine the \(x\)-coordinates by setting each factor to zero, resulting in \(x = -1\) and \(x = -3\).
  • Substitute these \(x\)-values into the linear equation \(y = 2x + 11\) to obtain the corresponding \(y\)-coordinates: \(y = 9\) for \(x = -1\), and \(y = 5\) for \(x = -3\).
  • Conclude that the intersection points of the system are \((-1, 9)\) and \((-3, 5)\).
  • Final Answer: \(\boxed{(-3, 5) \text{ and } (-1, 9)}\).

Explanation

  1. Analysis of the System of Equations

We are given a system of equations consisting of a quadratic equation \(y = -x^2 - 2x + 8\) and a linear equation \(y = 2x + 11\). The first three steps to solve this system are provided:

  1. Setting the expressions for \(y\) equal to each other: \(-x^2 - 2x + 8 = 2x + 11\).
  2. Simplifying the equation to the form \(0 = x^2 + 4x + 3\).
  3. Factoring the quadratic: \(0 = (x + 1)(x + 3)\).

Our goal is to find the corresponding \((x, y)\) coordinate pairs that satisfy these equations.

  1. Solving for x-coordinates

From Step 3, we have the factored equation \(0 = (x + 1)(x + 3)\). To find the values of \(x\) that satisfy the system, we set each factor equal to zero:

  • \(x + 1 = 0 \implies x = -1\)
  • \(x + 3 = 0 \implies x = -3\)

These are the two \(x\)-coordinates where the graphs of the two equations intersect.

  1. Finding the y-coordinates and Verifying Solutions

Now, we substitute each \(x\)-value into the linear equation \(y = 2x + 11\) to find the corresponding \(y\)-coordinates:

  • For \(x = -1\):

\(y = 2(-1) + 11 = -2 + 11 = 9\).
This gives the coordinate pair \((-1, 9)\).

  • For \(x = -3\):

\(y = 2(-3) + 11 = -6 + 11 = 5\).
This gives the coordinate pair \((-3, 5)\).

We can verify these points by plugging them into the original quadratic equation \(y = -x^2 - 2x + 8\):

  • For \((-1, 9)\): \(y = -(-1)^2 - 2(-1) + 8 = -1 + 2 + 8 = 9\). (Matches)
  • For \((-3, 5)\): \(y = -(-3)^2 - 2(-3) + 8 = -9 + 6 + 8 = 5\). (Matches)

Thus, the solutions to the system are \((-1, 9)\) and \((-3, 5)\).

Examples

In real-life scenarios, finding the intersection points of two equations is crucial for optimization and balancing. For instance, if one function represents the cost of producing a certain amount of goods and another represents the revenue earned from selling them, finding where these two functions are equal (the "break-even points") helps a business determine the range of production that leads to a profit. By solving the system, we identify exactly where the revenue and cost curves meet.

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