`8x+3y=25`
`4x+5y=23` then find `x:y`.

Options

A. `2 : 3`
B. `2 : 5`
C. `3 : 5`
D. `1 : 5`

Answer

To solve the equations \(8x + 3y = 25\) and \(4x + 5y = 23\) and find the ratio \(x:y\), follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \(8x + 3y = 25\) (Equation 1)
2. \(4x + 5y = 23\) (Equation 2)

### Step 2: Multiply Equation 2 by 2
To eliminate \(x\), we can multiply Equation 2 by 2:
\[ 2(4x + 5y) = 2(23) \]
This gives us:
\[ 8x + 10y = 46 \quad \text{(Equation 3)} \]

### Step 3: Subtract Equation 1 from Equation 3
Now, we will subtract Equation 1 from Equation 3:
\[ (8x + 10y) - (8x + 3y) = 46 - 25 \]
This simplifies to:
\[ 10y - 3y = 21 \]
Thus, we have:
\[ 7y = 21 \]

### Step 4: Solve for \(y\)
Now, divide both sides by 7:
\[ y = \frac{21}{7} = 3 \]

### Step 5: Substitute \(y\) back into Equation 1
Now that we have \(y\), we can substitute it back into Equation 1 to find \(x\):
\[ 8x + 3(3) = 25 \]
This simplifies to:
\[ 8x + 9 = 25 \]
Subtract 9 from both sides:
\[ 8x = 25 - 9 \]
\[ 8x = 16 \]
Now, divide by 8:
\[ x = \frac{16}{8} = 2 \]

### Step 6: Find the ratio \(x:y\)
Now that we have both \(x\) and \(y\):
- \(x = 2\)
- \(y = 3\)

The ratio \(x:y\) is:
\[ x:y = 2:3 \]

### Final Answer
Thus, the ratio \(x:y\) is \(2:3\).

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