Question

Use the graphing method to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&x+y=8\\\&x+y=-1\end{aligned} $$

Short answer

The system has 0 solutions as the given lines are parallel and do not intersect.

Step-by-step solution

Expression of the Equations in Slope-Intercept Form

First, need to write the given equations in the slope intercept form, \(y = mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. The provided equations are already in this format.\n\nEquation 1: \(y = -x + 8\)\n\nEquation 2: \(y = -x - 1\)

Graphing of the Equations

Each of these equations represent a line on a graph. Plot these to get a vivid image of what's happening. The graph will have two straight lines. The line \(y = -x + 8\) intercepts the y-axis at the point (0,8), and the line \(y = -x - 1\) intercepts the y-axis at the point (0,-1). It's clear that both lines are parallel and never meet.

Determining Number of Solutions

For a system of linear equations to have a solution, the lines represented by the equations must intersect at a point. In this case, the two lines are parallel and do not intersect, which means this system of equations does not have a solution. Therefore, the system has 0 solutions.