If `x + y -7 =0` and `3x + y -13 = 0`. then what is `4x^(2) + y^(2) + 4xy` equal to?

Options

A. 75
B. 85
C. 91
D. 100

Answer

To solve the problem step by step, we will first solve the system of linear equations given and then substitute the values of \(x\) and \(y\) into the expression \(4x^2 + y^2 + 4xy\).

### Step 1: Write down the equations
We have the following two equations:
1. \(x + y - 7 = 0\) (Equation 1)
2. \(3x + y - 13 = 0\) (Equation 2)

### Step 2: Solve for \(y\) in terms of \(x\)
From Equation 1, we can express \(y\) in terms of \(x\):
\[ y = 7 - x \]

### Step 3: Substitute \(y\) into Equation 2
Now, we will substitute \(y\) from Step 2 into Equation 2:
\[ 3x + (7 - x) - 13 = 0 \]
Simplifying this gives:
\[ 3x - x + 7 - 13 = 0 \]
\[ 2x - 6 = 0 \]

### Step 4: Solve for \(x\)
Now, we solve for \(x\):
\[ 2x = 6 \implies x = \frac{6}{2} = 3 \]

### Step 5: Substitute \(x\) back to find \(y\)
Now that we have \(x\), we can find \(y\):
\[ y = 7 - x = 7 - 3 = 4 \]

### Step 6: Substitute \(x\) and \(y\) into the expression
Now we substitute \(x = 3\) and \(y = 4\) into the expression \(4x^2 + y^2 + 4xy\):
\[ 4x^2 + y^2 + 4xy = 4(3^2) + (4^2) + 4(3)(4) \]
Calculating each term:
\[ = 4(9) + 16 + 4(12) \]
\[ = 36 + 16 + 48 \]

### Step 7: Combine the results
Now we add these values together:
\[ 36 + 16 + 48 = 100 \]

### Final Answer
Thus, the value of \(4x^2 + y^2 + 4xy\) is \(100\).

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