Question

solve the given system by the method of your choice.\

$$\begin{cases}5x + 3y = 0\\\\3x = 4y\\end{cases}$$

\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\a. there is exactly one solution. the solution set is \boxed{}.\\(\text{(simplify your answer. type an ordered pair.)}\\)\\b. there are infinitely many solutions. the solution set is \\(\\{(x,y)|5x + 3y = 0\\}\\) or \\(\\{(x,y)|3x = 4y\\}\\).\\c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Express y from the second equation

From \(3x = 4y\), we can solve for \(y\) as \(y=\frac{3x}{4}\).

Step2: Substitute y into the first equation

Substitute \(y = \frac{3x}{4}\) into \(5x + 3y = 0\). We get \(5x+3\times\frac{3x}{4}=0\).
Simplify the left - hand side: \(5x+\frac{9x}{4}=0\). Combine the terms: \(\frac{20x + 9x}{4}=0\), which is \(\frac{29x}{4}=0\).
Multiply both sides by \(\frac{4}{29}\), we find that \(x = 0\).

Step3: Find the value of y

Substitute \(x = 0\) into \(y=\frac{3x}{4}\), then \(y=\frac{3\times0}{4}=0\).

Answer:

\((0,0)\)