Question

what is the solution to the system of equations?
$-6x - \frac{2}{5}y = 8$
$\frac{1}{2}x + 3y = 29$
\\(\circ\\) \\((10, -2)\\)
\\(\circ\\) \\((10, 2)\\)
\\(\circ\\) \\((2, 10)\\)
\\(\circ\\) \\((-2, 10)\\)

Explanation:

Step1: Eliminate fractions in Eq1

Multiply Eq1 by 5:
$$-6x \times 5 - \frac{2}{5}y \times 5 = 8 \times 5$$
$$-30x - 2y = 40$$

Step2: Eliminate fractions in Eq2

Multiply Eq2 by 2:
$$\frac{1}{2}x \times 2 + 3y \times 2 = 29 \times 2$$
$$x + 6y = 58$$

Step3: Isolate $x$ from Eq2

Rearrange Eq2 to solve for $x$:
$$x = 58 - 6y$$

Step4: Substitute $x$ into modified Eq1

Substitute $x=58-6y$ into $-30x -2y=40$:
$$-30(58 - 6y) - 2y = 40$$
$$-1740 + 180y - 2y = 40$$
$$178y = 40 + 1740$$
$$178y = 1780$$
$$y = 10$$

Step5: Solve for $x$ using $y=10$

Substitute $y=10$ into $x=58-6y$:
$$x = 58 - 6 \times 10$$
$$x = 58 - 60$$
$$x = -2$$

Answer:

(-2, 10)