solve each equation for the indicated variable abc = 1/2; b
The correct answer and explanation is:
To solve the equation abc=12abc = \frac{1}{2} for the variable bb, follow these steps:
Step-by-step solution:
The given equation is: abc=12abc = \frac{1}{2}
To isolate bb, divide both sides of the equation by acac (assuming a≠0a \neq 0 and c≠0c \neq 0): b=12acb = \frac{1}{2ac}
So, the correct answer is: b=12ac\boxed{b = \frac{1}{2ac}}
300-word explanation:
This equation involves three variables: aa, bb, and cc, all multiplied together and set equal to one-half. The goal is to solve for the variable bb, which means expressing bb alone on one side of the equation and everything else on the other side.
In algebra, when multiple variables are multiplied together, isolating one of them requires dividing both sides of the equation by the other variables. Since abc=12abc = \frac{1}{2}, and aa and cc are both multiplied with bb, the inverse operation is to divide both sides of the equation by the product acac. This will cancel out aa and cc from the left-hand side, leaving bb by itself.
So, divide both sides of the equation by acac: abcac=12ac\frac{abc}{ac} = \frac{1}{2ac}
On the left side, the aa and cc cancel out: b=12acb = \frac{1}{2ac}
This shows that bb depends inversely on both aa and cc. If either aa or cc increases, the value of bb decreases, assuming the other variables remain constant. This solution also shows the importance of understanding algebraic manipulation, especially when isolating variables in equations that involve multiplication.
Understanding how to rearrange and solve for a specific variable is a fundamental algebra skill and is especially useful in fields like physics, economics, and engineering, where formulas with multiple variables are common.