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
