Solve the following system of linear equations by cross-miltiplication method.
`3x + 4y = 7 and 2x + 5y = 6`

Options

A. `(11)/(7) , (4)/(7)`
B. `- (11)/(7) , (4)/(7)`
C. `11, (5)/(7)`
D. None of these

Answer

To solve the given system of linear equations using the cross-multiplication method, we will follow these steps:

### Step 1: Write the equations in standard form
The given equations are:
1. \(3x + 4y = 7\)
2. \(2x + 5y = 6\)

We can rewrite these equations in the form \(Ax + By + C = 0\):
1. \(3x + 4y - 7 = 0\)
2. \(2x + 5y - 6 = 0\)

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- For the first equation \(3x + 4y - 7 = 0\):
- \(A_1 = 3\), \(B_1 = 4\), \(C_1 = -7\)

- For the second equation \(2x + 5y - 6 = 0\):
- \(A_2 = 2\), \(B_2 = 5\), \(C_2 = -6\)

### Step 3: Set up the cross-multiplication
Using the cross-multiplication method, we set up the following ratios:

\[ \frac{x}{A_1 B_2 - A_2 B_1} = \frac{y}{B_1 C_2 - B_2 C_1} = \frac{1}{C_1 A_2 - C_2 A_1} \]

### Step 4: Calculate the determinants
Now we calculate the determinants:

1. For \(A_1 B_2 - A_2 B_1\):
\[ 3 \cdot 5 - 2 \cdot 4 = 15 - 8 = 7 \]

2. For \(B_1 C_2 - B_2 C_1\):
\[ 4 \cdot (-6) - 5 \cdot (-7) = -24 + 35 = 11 \]

3. For \(C_1 A_2 - C_2 A_1\):
\[ -7 \cdot 2 - (-6) \cdot 3 = -14 + 18 = 4 \]

### Step 5: Write the ratios
Now we can write the ratios:
\[ \frac{x}{7} = \frac{y}{11} = \frac{1}{4} \]

### Step 6: Solve for \(x\) and \(y\)
From the ratios, we can express \(x\) and \(y\):
1. From \(\frac{x}{7} = \frac{1}{4}\):
\[ x = 7 \cdot \frac{1}{4} = \frac{7}{4} \]

2. From \(\frac{y}{11} = \frac{1}{4}\):
\[ y = 11 \cdot \frac{1}{4} = \frac{11}{4} \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = \frac{7}{4}, \quad y = \frac{11}{4} \]