Answer
$\left(-17, \frac{161}{2}\right)$ and $\left(1, \frac{17}{2}\right)$
Work Step by Step
We will use substitution to solve this system of equations. We substitute one of the expressions given for the $y$ term, which would mean that we are going to set the two equations equal to one another to solve for $x$ first:
$\frac{1}{2}x^2 + 4x + 4 = -4x + 12\frac{1}{2}$
We want to move all terms to the left side of the equation:
$\frac{1}{2}x^2 + 4x + 4x + 4 - 12\frac{1}{2} = 0$
Change constants into equivalent fractions:
$\frac{1}{2}x^2 + 4x + 4x + \frac{8}{2} - \frac{25}{2} = 0$
Combine like terms:
$\frac{1}{2}x^2 + 8x - \frac{17}{2} = 0$
Multiply both sides of the equation by $2$ to convert all fractions into whole numbers:
$x^2 + 16x - 17 = 0$
Factor the quadratic equation. The quadratic equation takes the form $ax^2 + bx + c = 0$. We need to find factors whose product is $ac$ but that sum up to $b$. In this exercise, $ac = -17$ and $b = 16$. The factors $17$ and $-1$ will work:
$(x + 17)(x - 1) = 0$
Set each factor equal to $0$.
First factor:
$x + 17 = 0$
Subtract $17$ from each side of the equation:
$x = -17$
Second factor:
$x - 1 = 0$
Add $1$ to each side of the equation:
$x = 1$
Now that we have the two possible values for $x$, we can plug them into one of the original equations to find the corresponding $y$ values. Let's use the first equation:
$y = \frac{1}{2}x^2 + 4x + 4$
Substitute the solution $-17$ for $x$:
$y = \frac{1}{2}(-17)^2 + 4(-17) + 4$
Evaluate exponents first:
$y = \frac{1}{2}(289) + 4(-17) + 4$
Multiply next:
$y = \frac{289}{2} - 68 + 4$
Convert all constants into equivalent fractions with the same denominator:
$y = \frac{289}{2} - \frac{136}{2} + \frac{8}{2}$
Add or subtract from left to right to solve:
$y = \frac{161}{2}$
Let's solve for $y$ using the other solution, $x = 1$:
$y = \frac{1}{2}(1)^2 + 4(1) + 4$
Evaluate exponents first:
$y = \frac{1}{2}(1) + 4(1) + 4$
Multiply next:
$y = \frac{1}{2} + 4 + 4$
Convert all constants into equivalent fractions with the same denominator:
$y = \frac{1}{2} + \frac{8}{2} + \frac{8}{2}$
Add or subtract from left to right to solve:
$y = \frac{17}{2}$
The solutions are $\left(-17, \frac{161}{2}\right)$ and $\left(1, \frac{17}{2}\right)$.
