Solve the following system of linear equations by cross-multiplication 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 system of linear equations using the cross-multiplication method, we have the following equations:

1. \(3x + 4y = 7\)
2. \(2x + 5y = 6\)

### Step 1: Rearranging the equations
We can rearrange both equations to set them equal to zero:

- From the first equation:
\[ 3x + 4y - 7 = 0 \]

- From the second equation:
\[ 2x + 5y - 6 = 0 \]

### Step 2: Identifying coefficients
Now, we identify the coefficients for the cross-multiplication method:

- For the first equation \(a_1 = 3\), \(b_1 = 4\), \(c_1 = -7\)
- For the second equation \(a_2 = 2\), \(b_2 = 5\), \(c_2 = -6\)

### Step 3: Setting up the cross-multiplication formula
The formulas for \(x\) and \(y\) using the cross-multiplication method are:

\[ x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1} \]

\[ y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1} \]

### Step 4: Calculating \(x\)
Now we substitute the values into the formula for \(x\):

\[ x = \frac{(4)(-6) - (5)(-7)}{(3)(5) - (2)(4)} \]

Calculating the numerator:

\[ 4 \times -6 = -24 \]
\[ 5 \times -7 = -35 \]
\[ -24 - (-35) = -24 + 35 = 11 \]

Calculating the denominator:

\[ 3 \times 5 = 15 \]
\[ 2 \times 4 = 8 \]
\[ 15 - 8 = 7 \]

So, we have:

\[ x = \frac{11}{7} \]

### Step 5: Calculating \(y\)
Now we substitute the values into the formula for \(y\):

\[ y = \frac{(-7)(2) - (-6)(3)}{(3)(5) - (2)(4)} \]

Calculating the numerator:

\[ -7 \times 2 = -14 \]
\[ -6 \times 3 = -18 \]
\[ -14 - (-18) = -14 + 18 = 4 \]

Using the same denominator as before:

\[ 15 - 8 = 7 \]

So, we have:

\[ y = \frac{4}{7} \]

### Final Solution
Thus, the solution to the system of equations is:

\[ x = \frac{11}{7}, \quad y = \frac{4}{7} \]