Solve the given system by the substitution method:
[tex]\[
\begin{array}{r}
4x + 3y = 0 \\
x - 4y = 0
\end{array}
\][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The solution set is [tex]\(\{ (\square, \square) \}\)[/tex]. (Type an ordered pair.)
B. There are infinitely many solutions.
C. There is no solution.



Answer :

To solve the given system of equations by the substitution method, follow these steps:

1. Given System of Equations:

[tex]\[ \begin{cases} 4x + 3y = 0 \quad \text{(Equation 1)} \\ x - 4y = 0 \quad \text{(Equation 2)} \end{cases} \][/tex]

2. Solve one equation for one variable:

Let's solve Equation 2 for [tex]\( x \)[/tex]:

[tex]\[ x - 4y = 0 \implies x = 4y \][/tex]

3. Substitute the expression for [tex]\( x \)[/tex] into the other equation:

Substitute [tex]\( x = 4y \)[/tex] into Equation 1:

[tex]\[ 4(4y) + 3y = 0 \][/tex]

4. Simplify and solve for [tex]\( y \)[/tex]:

[tex]\[ 16y + 3y = 0 \implies 19y = 0 \implies y = 0 \][/tex]

5. Substitute [tex]\( y \)[/tex] back into the expression for [tex]\( x \)[/tex]:

Substitute [tex]\( y = 0 \)[/tex] into [tex]\( x = 4y \)[/tex]:

[tex]\[ x = 4(0) = 0 \][/tex]

6. State the solution set:

The solution to the system is the ordered pair [tex]\((x, y) = (0, 0)\)[/tex].

Therefore, the correct choice is:

A. The solution set is [tex]\(\{(0, 0)\}\)[/tex].