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Solve for b in the literal equation $\\frac{1}{2}ab^2 + c = d$ with positive variables
Mathematics
Grade 9 (Junior High School)
Question Content
For all positive values of a, b, c, and d, when $\\frac{1}{2}ab^2 + c = d$, which of the following expressions is equal to b?
Correct Answer
G. $\\sqrt{\\frac{2(d-c)}{a}}$
Detailed Solution Steps
1
Step 1: Isolate the term containing $b^2$ by subtracting $c$ from both sides of the equation: $\\frac{1}{2}ab^2 = d - c$
2
Step 2: Eliminate the coefficient $\\frac{1}{2}$ by multiplying both sides of the equation by 2: $ab^2 = 2(d - c)$
3
Step 3: Isolate $b^2$ by dividing both sides of the equation by $a$ (since $a$ is positive, we don't have to worry about negative division affecting the equality): $b^2 = \\frac{2(d - c)}{a}$
4
Step 4: Solve for $b$ by taking the square root of both sides. Since $b$ is positive, we only take the positive square root: $b = \\sqrt{\\frac{2(d - c)}{a}}$
Knowledge Points Involved
1
Isolating a variable (Literal Equations)
Literal equations are equations with multiple variables, and isolating a variable means rearranging the equation to get that variable alone on one side. This uses inverse operations (subtraction for addition, multiplication for division, square roots for squares) to undo operations and solve for the target variable. It is used when you need to express one variable in terms of others, common in algebra and science formulas.
2
Inverse Operations
Inverse operations are pairs of operations that reverse each other's effects: addition and subtraction are inverses, multiplication and division are inverses, squaring and taking the square root are inverses. They are used to isolate variables in equations by undoing the operations applied to the target variable.
3
Square Roots of Positive Numbers
For any positive real number $x$, the positive square root $\\sqrt{x}$ is the positive number that when multiplied by itself equals $x$. Since the problem states all values of $a, b, c, d$ are positive, we only need to consider the positive square root when solving for $b$, as a negative value for $b$ would not fit the problem's constraints.
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