Question

which statement correctly describes the solution to this system of equations? \\(\

$$\begin{cases} -3x + 2y = -22 \\\\ -5y = -4x + 27 \\end{cases}$$

\\) \\(\bigcirc\\) a. there is no solution. \\(\bigcirc\\) b. there are infinitely many solutions. \\(\bigcirc\\) c. there is exactly one solution at \\((1, 8)\\). \\(\bigcirc\\) d. there is exactly one solution at \\((8, 1)\\).

Explanation:

Step1: Rewrite equations in standard form

First equation: \(-3x + 2y = -22\) (already in \(Ax + By = C\) form).
Second equation: \(-5y = -4x + 27\) → \(4x - 5y = 27\) (rearranged to standard form).

Step2: Solve using substitution or elimination

Let's use elimination. Multiply the first equation by 5 and the second by 2 to eliminate \(y\):
First equation (×5): \( -15x + 10y = -110 \)
Second equation (×2): \( 8x - 10y = 54 \)

Step3: Add the two equations

Adding them: \((-15x + 8x) + (10y - 10y) = -110 + 54\)
Simplify: \(-7x = -56\) → \(x = 8\) (divide both sides by -7).

Step4: Substitute \(x = 8\) into one equation

Use the first equation: \(-3(8) + 2y = -22\) → \(-24 + 2y = -22\)
Add 24 to both sides: \(2y = 2\) → \(y = 1\).

Step5: Check the solution

The solution is \((8, 1)\), so there is exactly one solution.

Answer:

D. There is exactly one solution at (8, 1).