Question

Graph and check to solve the linear system. $$ \begin{aligned} &x-y=1\\\ &5 x-4 y=0 \end{aligned} $$

Short answer

The short answer, or the point at which the lines intersect on the graph, will be the solution to the system. This will depend on the accurate drawing of the lines and the identification of the intersection point.

Step-by-step solution

Rearrange the Equations

First we need to rearrange both equations into slope-intercept form, \(y = mx + b\), so that we can graph them easily. For the first one, the equation \(x - y = 1\) can be rearranged as \(y = x - 1\). For the second equation, \(5x - 4y = 0\) can be rewritten as \(y = \frac{5}{4}x\).

Graph the Equations

With the equations in slope-intercept form, we can now plot them on a graph. Draw a vertical line (y-axis) and a horizontal line (x-axis). Now, for the first equation \(y = x - 1\), the y-intercept is -1 so place a point there. The slope is 1 which means for each step to the right on the x-axis, move one step up on the y-axis. For the second equation \(y = \frac{5}{4}x\), the y-intercept is 0 so place a point there, and then for every 4 steps to the right on the x-axis, move 5 steps up on the y-axis.

Find the Intersection Point

The solution to the system is where the two lines intersect each other on the graph. So by examining the graph, we try to find this intersection point which will be the values of \(x\) and \(y\) that satisfy both equations.