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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 11: Q. 200 (page 1147)

In the following exercises, solve the system of equations by using substitution.
$\{\begin{cases} x^{2} + 4 y^{2} = 4 \\ y = \frac{1}{2} x - 1 \end{cases}$

Short Answer

Expert verified

The solutions of the system are $(0, - 1), (2, 0)$ .

Step by step solution

01

Step 1. Given Information

The given system of equations is

$\{\begin{cases} x^{2} + 4 y^{2} = 4 \\ y = \frac{1}{2} x - 1 \end{cases}$

02

Step 2. Substitution

Substitute $y = \frac{1}{2} x - 1$ into $x^{2} + 4 y^{2} = 4$ and simplify.

$\begin{array}{rcl} x^{2} + 4 {(\frac{1}{2} x - 1)}^{2} & = & 4 \\ x^{2} + x^{2} + 4 - 4 x & = & 4 \\ 2 x^{2} - 4 x & = & 0 \\ 2 x (x - 2) & = & 0 \end{array}$

So, the value ofxis $x = 0, 2$ .

03

Step 3. Solve for y

Substitute $x = 0$ into $y = \frac{1}{2} x - 1$ and solve for y.

$\begin{array}{rcl} y & = & \frac{1}{2} 脳 0 - 1 \\ = & - 1 \end{array}$

Now, substitute $x = 2$ into $y = \frac{1}{2} x - 1$ and solve for y.

$\begin{array}{rcl} y & = & \frac{1}{2} 脳 2 - 1 \\ = & 1 - 1 \\ = & 0 \end{array}$

So, the solutions of the system are $(0, - 1), (2, 0)$ .

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