To solve the system of equations algebraically, we need to find the points where the two functions intersect. This involves setting the equations equal to each other and solving for the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
The system of equations is given by:
1. [tex]\( y = -\frac{1}{2}x - 7 \)[/tex]
2. [tex]\( y = x^2 + 6x + 5 \)[/tex]
First, we set the two equations equal to each other:
[tex]\[ -\frac{1}{2}x - 7 = x^2 + 6x + 5 \][/tex]
To clear the fraction, we can multiply every term by 2:
[tex]\[ -x - 14 = 2x^2 + 12x + 10 \][/tex]
Next, we rearrange the equation to move all terms to one side, resulting in a standard quadratic equation:
[tex]\[ 0 = 2x^2 + 12x + 10 + x + 14 \][/tex]
[tex]\[
0 = 2x^2 + 13x + 24
\][/tex]
Now, we solve for [tex]\(x\)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( b = 13 \)[/tex], and [tex]\( c = 24 \)[/tex]:
[tex]\[
b^2 - 4ac = 13^2 - 4 \cdot 2 \cdot 24 = 169 - 192 = -23
\][/tex]
Since the discriminant ([tex]\( -23 \)[/tex]) is negative, there are no real solutions for [tex]\(x\)[/tex]. This means that the parabolic equation [tex]\( y = x^2 + 6x + 5 \)[/tex] and the linear equation [tex]\( y = -\frac{1}{2} x - 7 \)[/tex] do not intersect in the real number plane.
Given that there are no real solutions, the system has complex solutions. These solutions are:
[tex]\[
x = -3.25 \pm 1.19895788082818i
\][/tex]
To find the corresponding [tex]\( y \)[/tex] values, we substitute these [tex]\( x \)[/tex] values back into the linear equation [tex]\( y = -\frac{1}{2}x - 7 \)[/tex]:
[tex]\[
y \approx -5.375 \pm 0.59947894041409i
\][/tex]
Thus, the solutions are:
[tex]\[
x_1 = -3.25 - 1.19895788082818i,
\][/tex]
[tex]\[
x_2 = -3.25 + 1.19895788082818i
\][/tex]
[tex]\[
y_1 = -5.375 + 0.59947894041409i,
\][/tex]
[tex]\[
y_2 = -5.375 - 0.59947894041409i
\][/tex]
Therefore, there are no real solutions, and the solutions can be summarized with complex numbers as:
[tex]\[
( (-3.25 - 1.19895788082818i, -3.25 + 1.19895788082818i), (-5.375 + 0.59947894041409i, -5.375 - 0.59947894041409i) )
\][/tex]
Since no real solutions exist, you are supposed to type "none" for the real solutions:
[tex]\[
\text{none}, \text{none}
\][/tex]
