Find the solution of linear equations `x+y=3` and `7x+6y=2`?

Options

A. -16, 19
B. 3, 2
C. 11, 9
D. -21, 20

Answer

To solve the linear equations \(x + y = 3\) and \(7x + 6y = 2\), we can follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(x + y = 3\) (Equation 1)
2. \(7x + 6y = 2\) (Equation 2)

### Step 2: Multiply Equation 1 by 7
To eliminate \(x\), we will multiply Equation 1 by 7:
\[ 7(x + y) = 7 \cdot 3 \]
This simplifies to:
\[ 7x + 7y = 21 \quad (Equation 3) \]

### Step 3: Write down Equation 2
We will keep Equation 2 as it is:
\[ 7x + 6y = 2 \quad (Equation 2) \]

### Step 4: Subtract Equation 2 from Equation 3
Now, we will subtract Equation 2 from Equation 3:
\[ (7x + 7y) - (7x + 6y) = 21 - 2 \]
This simplifies to:
\[ 7y - 6y = 19 \]
So, we have:
\[ y = 19 \]

### Step 5: Substitute the value of \(y\) back into Equation 1
Now that we have the value of \(y\), we will substitute it back into Equation 1 to find \(x\):
\[ x + 19 = 3 \]
To isolate \(x\), we subtract 19 from both sides:
\[ x = 3 - 19 \]
This simplifies to:
\[ x = -16 \]

### Step 6: Write the final solution
Thus, the solution to the system of equations is:
\[ x = -16, \quad y = 19 \]

### Summary
The solution to the equations \(x + y = 3\) and \(7x + 6y = 2\) is:
\[ (x, y) = (-16, 19) \]