Select the correct answer.

The difference between two numbers is 8. When twice the first number is added to three times the second number, the result is 51. What are the two numbers?

A. 12 and 4
B. 15 and 7
C. 20 and 12
D. 23 and 15



Answer :

Let's solve this step by step.

1. Identify the equations based on the given conditions:
- Let the two numbers be [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- The first condition states that the difference of the two numbers is 8. This gives us the equation:
[tex]\[ x - y = 8 \quad \text{(Equation 1)} \][/tex]
- The second condition states that twice the first number added to three times the second number equals 51. This gives us the equation:
[tex]\[ 2x + 3y = 51 \quad \text{(Equation 2)} \][/tex]

2. Solve the system of equations:
- From Equation 1, we can express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ x = y + 8 \][/tex]

- Substitute [tex]\( x \)[/tex] in Equation 2:
[tex]\[ 2(y + 8) + 3y = 51 \][/tex]
Simplify inside the parentheses and distribute:
[tex]\[ 2y + 16 + 3y = 51 \][/tex]
Combine like terms:
[tex]\[ 5y + 16 = 51 \][/tex]
Subtract 16 from both sides:
[tex]\[ 5y = 35 \][/tex]
Divide by 5:
[tex]\[ y = 7 \][/tex]

- Now, substitute the value of [tex]\( y \)[/tex] back into the expression for [tex]\( x \)[/tex]:
[tex]\[ x = y + 8 = 7 + 8 = 15 \][/tex]

3. Conclusion:
The two numbers that satisfy both conditions are [tex]\( 15 \)[/tex] and [tex]\( 7 \)[/tex].

So, the correct answer is:
O B. 15 and 7