Question

In the following exercises, solve the systems of equations by substitution.

$$\begin{gathered} 15x+4y=6 \\ -30x-8y=-12 \end{gathered}$$

Short answer

The equations have infinitely many solutions.

Step-by-step solution

Step 1. Given Information  

The given equations are

$$\begin{gathered} 15x+4y=6 \\ -30x-8y=-12 \end{gathered}$$

Step 2. Calculation   

Solve the second equation for variable y.

$$\begin{gathered} -30x-8y=-12 \\ -8y=-12+30x \\ y=\frac{-12+30x}{-8} \\ y=\frac{3}{2}-\frac{15}{4}x \end{gathered}$$

Substitute the obtained value of variable y in the first equation to find the value of variable x.

$$\begin{gathered} 15x+4y=6 \\ 15x+4(\frac{3}{2}-\frac{15}{4}x)=6 \\ 15x+6-15x=6 \\ 6=6 \end{gathered}$$

Since $6=6$is a true statement, the system is consistent. The equations are dependent. The graph of these two equations would give the same line. Thus the system of the linear equation has infinitely many solutions.