If `x/4+y/3=10/24 ` and `x/2+y=1`, then the value of (x+y) is

Options

A. 1
B. `3//2`
C. 2
D. 4

Answer

To solve the equations \( \frac{x}{4} + \frac{y}{3} = \frac{10}{24} \) and \( \frac{x}{2} + y = 1 \), we will follow these steps:

### Step 1: Simplify the first equation
The first equation is:
\[ \frac{x}{4} + \frac{y}{3} = \frac{10}{24} \]
We can simplify \( \frac{10}{24} \) to \( \frac{5}{12} \). So, we rewrite the equation as:
\[ \frac{x}{4} + \frac{y}{3} = \frac{5}{12} \]

### Step 2: Clear the fractions
To eliminate the fractions, we can multiply the entire equation by the least common multiple (LCM) of the denominators, which is 12:
\[ 12 \left(\frac{x}{4}\right) + 12 \left(\frac{y}{3}\right) = 12 \left(\frac{5}{12}\right) \]
This simplifies to:
\[ 3x + 4y = 5 \]
This is our first equation.

### Step 3: Simplify the second equation
The second equation is:
\[ \frac{x}{2} + y = 1 \]
To eliminate the fraction, we multiply the entire equation by 2:
\[ 2 \left(\frac{x}{2}\right) + 2y = 2 \cdot 1 \]
This simplifies to:
\[ x + 2y = 2 \]
This is our second equation.

### Step 4: Set up the system of equations
Now we have the following system of equations:
1. \( 3x + 4y = 5 \) (Equation 1)
2. \( x + 2y = 2 \) (Equation 2)

### Step 5: Use the elimination method
We can multiply Equation 2 by 3 to align the coefficients of \( x \):
\[ 3(x + 2y) = 3 \cdot 2 \]
This gives us:
\[ 3x + 6y = 6 \quad \text{(Equation 3)} \]

### Step 6: Subtract Equation 1 from Equation 3
Now we subtract Equation 1 from Equation 3:
\[ (3x + 6y) - (3x + 4y) = 6 - 5 \]
This simplifies to:
\[ 2y = 1 \]
Dividing both sides by 2 gives:
\[ y = \frac{1}{2} \]

### Step 7: Substitute \( y \) back to find \( x \)
Now we substitute \( y = \frac{1}{2} \) back into Equation 2:
\[ x + 2\left(\frac{1}{2}\right) = 2 \]
This simplifies to:
\[ x + 1 = 2 \]
Subtracting 1 from both sides gives:
\[ x = 1 \]

### Step 8: Find \( x + y \)
Now we can find \( x + y \):
\[ x + y = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]

### Final Answer
The value of \( x + y \) is:
\[ \frac{3}{2} \]

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