Question

solve the system of equations by any convenient method, if a solution exists.\

$$\begin{cases}3x - 4y = 22\\\\15x - 6y = 54\\end{cases}$$

\select the correct answer below and, if necessary, fill in any answer boxes to complete your choice.\\(\bigcirc\\) a. the solution is \\(x = \square\\) and \\(y = \square\\). (type integers or simplified fractions.)\\(\bigcirc\\) b. there are infinitely many solutions.\\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Eliminate x via scaling eq1

Multiply first equation by 5:
$$5(3x - 4y) = 5(22)$$
$$15x - 20y = 110$$

Step2: Subtract eq2 from new eq1

Subtract second equation to eliminate x:
$$(15x - 20y) - (15x - 6y) = 110 - 54$$
$$-14y = 56$$

Step3: Solve for y

Isolate y by dividing by -14:
$$y = \frac{56}{-14} = -4$$

Step4: Substitute y into original eq1

Plug y=-4 into $3x - 4y = 22$:
$$3x - 4(-4) = 22$$
$$3x + 16 = 22$$

Step5: Solve for x

Isolate x:
$$3x = 22 - 16 = 6$$
$$x = \frac{6}{3} = 2$$

Answer:

A. The solution is x = 2 and y = -4. (Type integers or simplified fractions.)