# How do you solve 5 = y - x and 4x^2 = -17x + y + 4?

Jan 29, 2017

I will let you finish the calculation

#### Explanation:

Given:

$5 = y - x \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . E q u a t i o n \left(1\right)$
$4 {x}^{2} = - 17 x + y + 4 \text{ } \ldots \ldots E q u a t i o n \left(2\right)$

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Consider equation(1)

Add $x$ to both sides.

$5 + x = y - x + x$
$5 + x = y \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots . E q u a t i o n \left({1}_{a}\right)$

Using $E q u a t i o n \left({1}_{a}\right)$ substitute for $\textcolor{red}{y}$ in $E q u a t i o n \left(2\right)$

$4 {x}^{2} = - 17 x + \textcolor{red}{y} + 4 \text{ "->" } 4 {x}^{2} = - 17 x + \textcolor{red}{5 + x} + 4$

$\text{ "->" } 4 {x}^{2} = - 16 x + 9$
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Now we have a quadratic equation

$4 {x}^{2} = - 16 x + 9 \text{ "->" } 4 {x}^{2} + 16 x - 9 = 0$

Use the formula to solve this:

Standard form $\to y - a {x}^{2} + b x + c$ where $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

I will let you do the next bit

You should get $x = - 2 \pm \frac{5}{2}$

By substitution you can then find the value of $y$

Jan 29, 2017

The parabola and the straight line meet at $\left(- \frac{9}{2} , \frac{1}{2}\right) \mathmr{and} \left(\frac{1}{2} , \frac{11}{2}\right)$
Illustrative Socratic graphs are inserted.

#### Explanation:

The first equation is of the form

y = a quadratic in x, and so, represents a parabola.

The line y = x+5 cuts the parabola, when

#4x^2+16x-9=0, giving x = -9/2 and 1/2.

Correspondingly,

y = x + 5 = 1/2 and 11/2.

So, the common points are $\left(\frac{1}{2} , \frac{11}{2}\right) \mathmr{and} \left(- \frac{9}{2} , \frac{1}{2}\right)$.

graph{(y-4x^2-17x+4)(y-x-5)=0 [-50, 50, -25, 25]}

graph{(y-4x^2-17x+4)(y-x-5)=0 [-20, 20, -10, 10]}