Question Find the solution of the system of equations. $$ 10 x-9 y=-32 $$ $$ 2 x-10 y=10 $$

Question

UpStudy AI · Accepted Answer

We start with the system of equations:

$$
10x - 9y = -32 \tag{1}
$$

$$
2x - 10y = 10 \tag{2}
$$

**Step 1.** Multiply equation $$(2)$$ by $$5$$ to match the coefficient of $$x$$ in equation $$(1)$$:

$$
5 \times (2x - 10y) = 5 \times 10
$$

which gives:

$$
10x - 50y = 50 \tag{3}
$$

**Step 2.** Subtract equation $$(1)$$ from equation $$(3)$$ to eliminate $$x$$:

$$
(10x - 50y) - (10x - 9y) = 50 - (-32)
$$

Simplify the left-hand side:

$$
10x - 50y - 10x + 9y = -41y
$$

Simplify the right-hand side:

$$
50 + 32 = 82
$$

So we have:

$$
-41y = 82
$$

**Step 3.** Solve for $$y$$:

$$
y = \frac{82}{-41} = -2
$$

**Step 4.** Substitute $$y = -2$$ back into equation $$(2)$$:

$$
2x - 10(-2) = 10
$$

Simplify:

$$
2x + 20 = 10
$$

Subtract $$20$$ from both sides:

$$
2x = -10
$$

Divide by $$2$$:

$$
x = -5
$$

Thus, the solution to the system is:

$$
(x, y) = (-5, -2)
$$
The solution to the system of equations is $$ x = -5 $$ and $$ y = -2 $$.

Question Find the solution of the system of equations. \[ 10 x-9 y=-32 \] \[ 2 x-10 y=10 \]