Chapter 11.6, Problem 15AYU

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.

$\{\begin{array}{l}y\text{ = }3x-\text{ 5}\\ x^2\text{ + }{y}^2\text{ = }5\end{array}$

To determine

To graph: The equation in the system and solve to find the points of intersection.

Answer to Problem 15AYU

Explanation of Solution

Given:

$y = 3x - 5$    -----Eq 1

$x^2 + y^2 = 5$    -----Eq 2

Graph:

Interpretation:

Solving for $x$ and $y$ from Eq 1 and Eq2:

Substituting Eq 1in Eq 2,

$x^2 + {\left(3x - 5\right)}^2 = 5$

$= > 10x^2 - 30x + 25 = 5$

$= > x^2 - 3x + 2 = 0$

Solving the above quadratic equation, $x = 1$ and $x = 2$

For $x = 1$ we get $y = \text{-}2$ and

For $x = 2$ we get $y = 1$.

The solution to the set of equation is $(1, \text{-}2)$ and $(2, 1)$.

The same can be seen from the graph where the equations $y = 3x - 5$ and $x^2 + y^2 = 5$ intersect at $(1, \text{-}2)$ and $(2, 1)$.