Question

Use the graphing method to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&\frac{3}{4} x+\frac{1}{2} y=10\\\&-\frac{3}{2} x-y=4\end{aligned} $$

Short answer

The system of equations has one solution, which is (4,8).

Step-by-step solution

Rewrite the Equations in Slope-Intercept Form

First, transform each equation into the slope-intercept form \(y = mx + b\), where m is the slope, and b is the y-intercept. The first equation becomes \(y = - \frac{3}{2}x + 20\) and the second equation simplifies to \(y = \frac{3}{2}x - 4\).

Graph the Equations

Next, we will graph these two lines. The first line (-3/2, 20) connects points at (0,20) and (13.33,0). The second line (3/2, -4) connects points at (0,-4) and (2.67,0). If done correctly, the two lines should intersect corresponding to the solution of the system.

Identify the Point of Intersection

After drawing, it can be observed that the lines intersect at the point (4,8). This means that \(x = 4\) and \(y = 8\) is the solution of the system.

Determine the Number of Solutions

As there exists a unique point of intersection, it can be concluded that the system of equations has one solution.