Question

Use a graph to solve the linear system. Check your solution algebraically. (Review 7.1 ) $$\begin{aligned}&4 x+5 y=20\\\&\frac{5}{4} x+y=4\end{aligned}$$

Short answer

The point of intersection - that is, the solution of given linear system - is (2,2).

Step-by-step solution

- Convert both equations into slope-intercept form

The given system of equations is in the standard form. In order to graph, it is easier if the equations are written in slope-intercept form (y = mx + b). This can be done by isolating 'y' on one side of the equation. Doing this for each equation, the converted equations are: \(y = - \frac{4}{5}x + 4\) and \(y = -\frac{5}{4}x + 4\).

- Graph the functions

Plot these two functions on the same set of axes. The x-intercepts (when y=0) for the two functions are found by setting y = 0 and solving for x. This gives \(x = 4\) for the first equation and \(x = \frac{16}{5}\) for the second equation. The y-intercepts are both \(4\) as visible even from the slope-intercept forms. Both lines have negative slopes but are not the same, so they will intersect at some point.

- Find the point of intersection

The point of intersection of the two lines represents the solution to the system of equations. If we find it graphically, we can see the two lines cross each other at the point (2,2). This is the solution.

- Verify the solution

Substitute the point (2,2) into both original equations to verify they are both true. For first equation , 4(2) + 5(2) equals to 20. And for second equation, \(\frac{5}{4}\)(2) + 2 equals to 4. This means that the solution is correct. So, the solution to the system is (2,2).