If `3x+5y=15 and x-2y=10`, what is the value of `2x+7y`?

Options

A. `5`
B. `10`
C. `15`
D. `25`

Answer

To solve the system of equations given by \(3x + 5y = 15\) and \(x - 2y = 10\) and find the value of \(2x + 7y\), we can follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(3x + 5y = 15\) (Equation 1)
2. \(x - 2y = 10\) (Equation 2)

### Step 2: Eliminate one variable
To eliminate \(x\), we can multiply Equation 2 by 3 so that the coefficients of \(x\) in both equations will be the same.

Multiply Equation 2 by 3:
\[ 3(x - 2y) = 3(10) \implies 3x - 6y = 30 \quad \text{(Equation 3)} \]

### Step 3: Subtract the equations
Now we will subtract Equation 3 from Equation 1:
\[ (3x + 5y) - (3x - 6y) = 15 - 30 \]
This simplifies to:
\[ 3x + 5y - 3x + 6y = -15 \]
\[ 11y = -15 \]
Now, solve for \(y\):
\[ y = -\frac{15}{11} \]

### Step 4: Substitute \(y\) back to find \(x\)
Now we will substitute the value of \(y\) back into Equation 2 to find \(x\):
\[ x - 2\left(-\frac{15}{11}\right) = 10 \]
This simplifies to:
\[ x + \frac{30}{11} = 10 \]
Subtract \(\frac{30}{11}\) from both sides:
\[ x = 10 - \frac{30}{11} \]
Convert 10 to a fraction with a denominator of 11:
\[ x = \frac{110}{11} - \frac{30}{11} = \frac{80}{11} \]

### Step 5: Calculate \(2x + 7y\)
Now we can find \(2x + 7y\):
\[ 2x + 7y = 2\left(\frac{80}{11}\right) + 7\left(-\frac{15}{11}\right) \]
This simplifies to:
\[ = \frac{160}{11} - \frac{105}{11} = \frac{55}{11} = 5 \]

### Final Answer
Thus, the value of \(2x + 7y\) is \(5\).

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