Question

solve the system of equations by the addition method. \\(\

$$\begin{cases}-3x + 5y = 47\\\\-5x - 3y = -1\\end{cases}$$

\\) select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\boldsymbol{\circ}\\) a. the solution is \\(\square\\). (simplify your answer. type an ordered pair.) \\(\boldsymbol{\circ}\\) b. there are infinitely many solutions. \\(\boldsymbol{\circ}\\) c. there is no solution.

Explanation:

Step1: Eliminate a variable (e.g., x)

Multiply the first equation by 5 and the second by 3 to make coefficients of x opposites.
First equation: \( 5(-3x + 5y) = 5(47) \) → \( -15x + 25y = 235 \)
Second equation: \( 3(-5x - 3y) = 3(-1) \) → \( -15x - 9y = -3 \)

Step2: Subtract the equations to eliminate x

\( (-15x + 25y) - (-15x - 9y) = 235 - (-3) \)
\( -15x + 25y + 15x + 9y = 238 \)
\( 34y = 238 \)

Step3: Solve for y

\( y = \frac{238}{34} = 7 \)

Step4: Substitute y = 7 into one equation (e.g., first)

\( -3x + 5(7) = 47 \)
\( -3x + 35 = 47 \)
\( -3x = 12 \)
\( x = -4 \)

Answer:

\((-4, 7)\)