Which shows the correct solution of the equation
1/2 a + 2/3b = 50 when b=30
The Correct Answer and Explanation is :
The correct answer is: ( a = 60 )
To solve the equation ( \frac{1}{2} a + \frac{2}{3} b = 50 ) when ( b = 30 ), we first substitute ( b ) with ( 30 ):
[
\frac{1}{2} a + \frac{2}{3}(30) = 50
]
Calculating ( \frac{2}{3}(30) ):
[
\frac{2}{3} \times 30 = 20
]
Now, substitute this value back into the equation:
[
\frac{1}{2} a + 20 = 50
]
Next, we need to isolate ( \frac{1}{2} a ) by subtracting ( 20 ) from both sides:
[
\frac{1}{2} a = 50 – 20
]
[
\frac{1}{2} a = 30
]
To find ( a ), multiply both sides of the equation by ( 2 ):
[
a = 30 \times 2
]
[
a = 60
]
Thus, the solution for ( a ) when ( b = 30 ) is ( a = 60 ).
Explanation
The process of solving this equation involves substitution and basic algebraic manipulation. Initially, the equation includes two variables, ( a ) and ( b ), which represent unknown quantities. However, by substituting the known value of ( b ) (which is ( 30 )), we simplify the equation to one variable.
After substituting ( b ) into the equation, we performed multiplication to find the value contributed by ( b ) in the equation. Then, we isolated the term containing ( a ) on one side of the equation by subtracting the known value from both sides. This step is crucial in algebra, as it allows us to simplify the equation further and focus on solving for ( a ).
Finally, we multiplied to eliminate the fraction, arriving at the final value of ( a ). This step-by-step approach showcases how algebra can be utilized to find unknown values systematically, emphasizing the importance of understanding each operation’s impact on the equation’s balance. The conclusion that ( a = 60 ) illustrates the successful application of algebraic principles in solving for unknowns.