The soution set of `4x - 3y = 10 and 3x + y=14` is

Options

A. `{(5,3)}`
B. `{(4,2)}`
C. `{(11,-1)}`
D. `{(2,3)}`

Answer

To find the solution set of the equations \(4x - 3y = 10\) and \(3x + y = 14\), we can solve these equations step by step.

### Step 1: Write down the equations
We have:
1. \(4x - 3y = 10\) (Equation 1)
2. \(3x + y = 14\) (Equation 2)

### Step 2: Manipulate Equation 2
To eliminate \(y\), we can manipulate Equation 2. We will multiply the entire Equation 2 by 3 to match the coefficient of \(y\) in Equation 1.

\[ 3(3x + y) = 3(14) \]
This simplifies to:
\[ 9x + 3y = 42 \quad \text{(Equation 3)} \]

### Step 3: Add Equation 1 and Equation 3
Now we will add Equation 1 and Equation 3 together:

\[ (4x - 3y) + (9x + 3y) = 10 + 42 \]

### Step 4: Simplify the equation
On the left side, \(-3y + 3y\) cancels out, so we have:

\[ 4x + 9x = 13x \]

On the right side, we have:

\[ 10 + 42 = 52 \]

Thus, we get:

\[ 13x = 52 \]

### Step 5: Solve for \(x\)
Now, we can solve for \(x\):

\[ x = \frac{52}{13} = 4 \]

### Step 6: Substitute \(x\) back into Equation 2
Next, we substitute \(x = 4\) back into Equation 2 to find \(y\):

\[ 3(4) + y = 14 \]

This simplifies to:

\[ 12 + y = 14 \]

### Step 7: Solve for \(y\)
Now, we solve for \(y\):

\[ y = 14 - 12 = 2 \]

### Step 8: Write the solution set
The solution set is:

\[ (x, y) = (4, 2) \]

### Conclusion
Thus, the solution set of the equations \(4x - 3y = 10\) and \(3x + y = 14\) is \((4, 2)\).

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