$$ x+2 y=4 $$ and $$ 2 x-y-3=0 $$ $$ 2 y-5 x=7 $$ and $$ 4 x+3 y+24=0 $$

Question

UpStudy AI · Accepted Answer

**Solving the first system**

We have:
$$
x+2y=4 \quad \text{(1)}
$$
$$
2x-y-3=0 \quad \text{(2)}
$$

1. Rewrite equation (2) as:
   $$
   2x-y=3 \quad \text{(2')}
   $$

2. Solve (2') for $$y$$:
   $$
   y=2x-3
   $$

3. Substitute $$y=2x-3$$ into (1):
   $$
   x+2(2x-3)=4
   $$
   $$
   x+4x-6=4
   $$
   $$
   5x-6=4
   $$

4. Solve for $$x$$:
   $$
   5x=10 \quad \Rightarrow \quad x=2
   $$

5. Substitute $$x=2$$ back into $$y=2x-3$$:
   $$
   y=2(2)-3=4-3=1
   $$

Thus, the solution for the first system is:
$$
\boxed{(x,y)=(2,1)}
$$

**Solving the second system**

We have:
$$
2y-5x=7 \quad \text{(3)}
$$
$$
4x+3y+24=0 \quad \text{(4)}
$$

1. Rewrite equation (4) as:
   $$
   4x+3y=-24
   $$

2. Solve (3) for $$y$$:
   $$
   2y=5x+7 \quad \Rightarrow \quad y=\frac{5x+7}{2}
   $$

3. Substitute $$y=\frac{5x+7}{2}$$ into equation (4):
   $$
   4x+3\left(\frac{5x+7}{2}\right)=-24
   $$

4. Multiply the entire equation by 2 to eliminate the fraction:
   $$
   8x+3(5x+7)=-48
   $$
   $$
   8x+15x+21=-48
   $$
   $$
   23x+21=-48
   $$

5. Solve for $$x$$:
   $$
   23x=-48-21 = -69 \quad \Rightarrow \quad x=-3
   $$

6. Substitute $$x=-3$$ back into $$y=\frac{5x+7}{2}$$:
   $$
   y=\frac{5(-3)+7}{2}=\frac{-15+7}{2}=\frac{-8}{2}=-4
   $$

Thus, the solution for the second system is:
$$
\boxed{(x,y)=(-3,-4)}
$$
The solutions are: 1. $$ (x, y) = (2, 1) $$ 2. $$ (x, y) = (-3, -4) $$

\( x+2 y=4 \) and \( 2 x-y-3=0 \) \( 2 y-5 x=7 \) and \( 4 x+3 y+24=0 \)