Certainly! Let's solve the given system of simultaneous equations step-by-step:
Given equations:
[tex]\[
\begin{cases}
2x + 6y = 0 \quad \text{(1)}\\
3x - 5y = -7 \quad \text{(2)}
\end{cases}
\][/tex]
Step 1: Solve Equation (1) for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
[tex]\[ 2x + 6y = 0 \][/tex]
Subtract [tex]\( 6y \)[/tex] from both sides:
[tex]\[ 2x = -6y \][/tex]
Divide both sides by 2:
[tex]\[ x = -3y \quad \text{(3)} \][/tex]
Step 2: Substitute Equation (3) into Equation (2).
Replace [tex]\( x \)[/tex] in Equation (2):
[tex]\[ 3(-3y) - 5y = -7 \][/tex]
Simplify:
[tex]\[ -9y - 5y = -7 \][/tex]
Combine like terms:
[tex]\[ -14y = -7 \][/tex]
Step 3: Solve for [tex]\( y \)[/tex].
Divide both sides by [tex]\(-14\)[/tex]:
[tex]\[ y = \frac{-7}{-14} \][/tex]
Simplify:
[tex]\[ y = \frac{1}{2} \][/tex]
Step 4: Substitute [tex]\( y = \frac{1}{2} \)[/tex] back into Equation (3) to solve for [tex]\( x \)[/tex].
[tex]\[ x = -3 \left(\frac{1}{2}\right) \][/tex]
Simplify:
[tex]\[ x = -\frac{3}{2} \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ x = -\frac{3}{2} \][/tex]
[tex]\[ y = \frac{1}{2} \][/tex]
So the answer is:
[tex]\[ \left( x, y \right) = \left( -\frac{3}{2}, \frac{1}{2} \right) \][/tex]
That's the detailed step-by-step solution for solving these simultaneous equations.
