Question

find the solution or solutions that satisfy this system of equations. if none of the points are solutions to the system, choose no solution. if the system has infinitely many solutions, choose infinitely many solutions.
draw
(1,6) (0, -5)
(3, -1) (5,0)
no solution infinitely many solutions

Explanation:

Step1: Analyze the two lines

The system of equations is represented by two lines on the graph. The solution to a system of linear equations is the point where the two lines intersect.

Step2: Find the intersection point

By looking at the graph, we can see that the two lines intersect at the point \((3, -1)\)? Wait, no, let's re - examine. Wait, let's find the equations of the lines.

First line: Let's take two points. For the blue line, it passes through \((0, - 2)\) and \((6,0)\). The slope \(m_1=\frac{0 - (-2)}{6-0}=\frac{2}{6}=\frac{1}{3}\). Using the slope - intercept form \(y = mx + b\), with \(b=-2\), the equation is \(y=\frac{1}{3}x - 2\).

Second line: It passes through \((0,-4)\) and \((3, - 1)\)? Wait, when \(x = 3\), let's check the first line: \(y=\frac{1}{3}(3)-2=1 - 2=-1\). And for the second line, let's find its slope. It passes through \((0,-4)\) and \((3, - 1)\), slope \(m_2=\frac{-1-(-4)}{3 - 0}=\frac{3}{3}=1\). Equation: \(y=x - 4\).

Now, let's find the intersection point by solving the two equations:

\(\frac{1}{3}x-2=x - 4\)

\(-2 + 4=x-\frac{1}{3}x\)

\(2=\frac{2}{3}x\)

\(x = 3\)

Substitute \(x = 3\) into \(y=x - 4\), we get \(y=3 - 4=-1\). So the intersection point is \((3,-1)\).

Answer:

\((3, - 1)\)