Problem 55 - Algebra for College Students

Question

Solve each equation. $$\frac{w}{6}=\frac{3}{2 w}$$

Step-by-Step Solution

Answer

w = \pm 3

Step 1: Cross Multiply

To eliminate the fractions, cross multiply the given equation: \(\frac{w}{6} = \frac{3}{2w}\)This becomes: \(w \times 2w = 6 \times 3\)

Step 2: Simplify the Equation

Simplify the equation obtained from cross multiplication:\(2w^2 = 18\)

Step 3: Solve for w

Divide both sides of the equation by 2 to isolate the variable:\(w^2 = 9\)Take the square root of both sides to find the value of \w:\(w = \pm 3\)

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations involving fractions. When you have an equation like \[\frac{a}{b} = \frac{c}{d}\], you can eliminate the fractions by cross multiplying. Cross multiplying means you multiply the numerator of one fraction by the denominator of the other fraction and set the two products equal. For example, in the equation \[\frac{w}{6} = \frac{3}{2w}\], cross multiplying gives you: \[w \times 2w = 6 \times 3.\] This technique simplifies the equation and makes it easier to solve.

Simplifying Equations

Simplifying equations involves combining like terms and reducing the equation to its simplest form. After cross multiplying in our example, we have: \[w \times 2w = 12.\] We then simplify this equation to make it easier to solve. In this case, combine the terms: \[2w^2 = 18.\] This step is crucial because a simplified equation is much easier to work with as you move forward to solving for the variable.

Solving for Variables

Solving for variables means finding the value(s) of the variable that make the equation true. In our example, after simplifying, we have: \[2w^2 = 18.\] To solve for the variable w, we need to isolate it. We do this by dividing both sides of the equation by 2: \[w^2 = 9.\] Now that we have isolated w squared, the next step will be to take the square root of both sides to find the value of w.

Square Roots

Square roots are used to find a number that, when multiplied by itself, gives the original number. In our example, we have: \[w^2 = 9.\] To solve for w, we take the square root of both sides. The square root of 9 is \(\pm 3\), because both 3 and -3, when squared, equal 9. Therefore, the solutions to the original equation are: \[w = 3\] and \[w = -3.\] Taking square roots is an essential step in solving quadratic equations and finding the values of variables.