Twice the larger of two numbers is three more than five times the smaller and the sum of four times the larger and three times the smaller is 71. What are the numbers?

Options

A. 43,8
B. 11,9
C. 14,5
D. 17,1

Answer

To solve the problem, we need to set up equations based on the information given in the question. Let's break it down step by step.

### Step 1: Define the Variables
Let:
- \( x \) = the larger number
- \( y \) = the smaller number

### Step 2: Set Up the Equations
From the problem statement, we can derive two equations:

1. **First Equation**: "Twice the larger of two numbers is three more than five times the smaller."
\[ 2x = 5y + 3 \]

2. **Second Equation**: "The sum of four times the larger and three times the smaller is 71."
\[ 4x + 3y = 71 \]

### Step 3: Solve the First Equation for \( x \)
We can rearrange the first equation to express \( x \) in terms of \( y \):
\[ 2x = 5y + 3 \implies x = \frac{5y + 3}{2} \]

### Step 4: Substitute \( x \) in the Second Equation
Now we substitute \( x \) from the first equation into the second equation:
\[ 4\left(\frac{5y + 3}{2}\right) + 3y = 71 \]
Multiply through by 2 to eliminate the fraction:
\[ 4(5y + 3) + 6y = 142 \]
This simplifies to:
\[ 20y + 12 + 6y = 142 \]
Combine like terms:
\[ 26y + 12 = 142 \]

### Step 5: Solve for \( y \)
Subtract 12 from both sides:
\[ 26y = 130 \]
Divide by 26:
\[ y = 5 \]

### Step 6: Substitute \( y \) Back to Find \( x \)
Now that we have \( y \), we substitute it back into the equation for \( x \):
\[ x = \frac{5(5) + 3}{2} = \frac{25 + 3}{2} = \frac{28}{2} = 14 \]

### Step 7: Conclusion
The two numbers are:
- Larger number \( x = 14 \)
- Smaller number \( y = 5 \)

### Final Answer
The numbers are \( 14 \) and \( 5 \).

---