Question

Type the correct answer in each box. Use numerals instead of words. Consider the systems of equations below. System A System B System C I2 + y2 = 17 y = 12 - 70 + 10 = y = – 2229 - 1 2 3 - y = -65 + 5 &r - y = -17 Determine the number of real solutions for each system of equations. System A has real solutions. System B has real solutions. System C has real solutions.

Answer 1

A) Given:

$\begin{gathered} x^2+y^2=17\ldots\ldots\ldots\text{.}(1) \\ y=-\frac{1}{2}x\ldots\ldots\ldots\ldots(2) \end{gathered}$

To find: The number of real solutions

Explanation:

Substitute equation (2) in (1), we get

$\begin{gathered} x^2+(-\frac{1}{2}x)^2=17 \\ x^2+\frac{x^2}{4}=17 \\ \frac{5x^2}{4}=17^{} \\ x^2=\frac{68}{5} \\ x^2-\frac{68}{5}=0\ldots\ldots.(3) \end{gathered}$

Here,

$a=1,b=0,\text{ and c=-}\frac{68}{5}$

So, the discriminant is,

$\begin{gathered} \Delta=b^2-4ac \\ =0-4(1)(-\frac{68}{5}) \\ =54.4 \\ >0 \end{gathered}$

Since the discriminant is greater than zero.

Hence, it has two real solutions

Final answer:

System A has two real solutions.