Question

use a graphing utility to graph the two equations in the system in the same viewing rectangle. then use the intersection feature to determine the solutions. { y = 3 x - 15 x ^ { 2 } + y ^ { 2 } = 145 } which of the following shows the graph of the equations? note that the viewing rectangle is 13 by 13. a. b. c. d.

Explanation:

Step1: Analyze the line equation \(y = 3x-15\)

The \(y -\)intercept of the line \(y = 3x - 15\) is \(- 15\). When \(x = 0\), \(y=-15\). When \(y = 0\), \(0=3x - 15\), then \(x = 5\).

Step2: Analyze the circle equation \(x^{2}+y^{2}=145\)

The standard form of a circle is \((x - a)^{2}+(y - b)^{2}=r^{2}\), where \((a,b)\) is the center and \(r\) is the radius. For \(x^{2}+y^{2}=145\), the center is \((0,0)\) and \(r=\sqrt{145}\approx12.04\).

Step3: Check the intersection points

Substitute \(y = 3x - 15\) into \(x^{2}+y^{2}=145\). We get \(x^{2}+(3x - 15)^{2}=145\). Expand: \(x^{2}+9x^{2}-90x + 225=145\). Combine like - terms: \(10x^{2}-90x + 80 = 0\). Divide by \(10\): \(x^{2}-9x + 8=0\). Factor: \((x - 1)(x - 8)=0\). So \(x = 1\) or \(x = 8\). When \(x = 1\), \(y=3\times1-15=-12\); when \(x = 8\), \(y=3\times8-15 = 9\).

Step4: Analyze the graph

The line \(y = 3x-15\) should pass through \((5,0)\) (x - intercept) and \((0,-15)\) (y - intercept) and intersect the circle \(x^{2}+y^{2}=145\) at \((1,-12)\) and \((8,9)\).

Answer:

A.