Mathematics Notes - Analytic Geometry Problems and Solutions

Find the point of intersection of the curve $x^2 +y^2 = 25$ and $3x + y =5$. Solution:

Graph the equations to estimate the intersection.

$x^2 +y^2 = 25\) and \(3x + y =5$

Isolate $y$ from $3x+y = 5$

$y = 5-3x$

Substitute this value to the equation of the circle to solve $x$

$x^2 +(5-3x)^2 = 25$

$x^2 + 25 -30x +9x^2 = 25$

$x^2 -30x +9x^2 =0$

$10x^2 -30x = 0$

$x = 3$ and $x = 0$

$y = 5-3(0) = 5$

$y = 5-3(3) = -4$

The point of intersections are @ $(0,5)$ and (3,-4)