Question

By inspecting the equations, what can you do to determine about the solution(s) of this system. 6y=12x+36 15y=45x+6Choose the correct answer A. The system has infinitely many solutions.B. The system has exactly one solution.C. The system has no solution.

Answer 1

We have the following system of equations:

$\begin{gathered} 6y=12x+36\text{ (1)} \\ 15y=45x+6\text{ (2)} \end{gathered}$

To determine if it has one solution, infinitely many solutions or no solution, let's rewrite the equations in slope-intercept form:

$\begin{gathered} y=mx+b \\ \text{where,} \\ m=\text{slope} \\ b=y-\text{intercept} \end{gathered}$$\begin{gathered} y=2x+6\text{ (1)} \\ y=3x+\frac{2}{5} \end{gathered}$

*If the slopes are the same but the y-intercepts are different, the system has no solution.

*If the slopes are different, the system has one solution.

*If the slopes are the same and the y-intercepts are the same, the system has many solutions.

Then, in this case since their slopes are different and their y-intercept are different, the system of equations has exactly one solution.