Problem 41 - Algebra for College Students

Question

Find the solution set to each equation. $$\frac{3 w}{3 w-5}=\frac{w}{w+2}$$

Step-by-Step Solution

Answer

The solution set is \{0\}.

Step 1 – Cross Multiply

Cross multiply the fractions to eliminate the denominators. This gives: \[ 3w(w + 2) = w(3w - 5) \]

Step 2 – Distribute and Expand

Next, distribute and expand both sides of the equation: \[ 3w^2 + 6w = 3w^2 - 5w \]

Step 3 – Move All Terms to One Side

Subtract \(3w^2\) from both sides to simplify the equation: \[ 6w = -5w \]

Step 4 – Combine Like Terms

Add \(5w\) to both sides to further simplify: \[ 11w = 0 \]

Step 5 – Solve for w

Solve for \(w\) by dividing both sides by 11: \[ w = 0 \]

Step 6 – Verify Solutions

Verify the solution by substituting \(w = 0\) back into the original equation: \[ \frac{3(0)}{3(0)-5}=\frac{0}{0+2} \] Simplifies to: \[ 0 = 0 \] Which is true, so the solution is verified.

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations with fractions. It involves multiplying the numerator of each fraction by the denominator of the other fraction. For example, if we have an equation \(\frac{a}{b} = \frac{c}{d}\), we cross multiply to get \[ a \times d = b \times c \]. This helps to eliminate the fractions and results in a simpler equation that can be solved more easily.

Distributive Property

The distributive property is used to multiply a single term by each term within a parenthesis. It follows the rule \[ a(b + c) = ab + ac \]. In our example, for the equation \[3w(w + 2) = w(3w - 5)\], we distribute 3w and w respectively to get \[ 3w^2 + 6w \] and \[ 3w^2 - 5w \]. This step helps to expand and simplify the equation.

Combining Like Terms

After distributing and expanding, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same power. For instance, in the equation \[3w^2 + 6w = 3w^2 - 5w\], the like terms are \[3w^2\] and \[3w^2\] as well as \[6w\] and \[-5w\]. When we subtract \[3w^2\] from both sides we get \[6w = -5w\]. We then add \[-5w\] to both sides to combine the terms, resulting in \[11w = 0\].

Solution Verification

Solution verification ensures that the values obtained actually satisfy the original equation. Plugging the solution back into the equation checks for any errors made during calculations. For instance, we substitute \[w = 0\] back into the original equation \[ \frac{3w}{3w-5} = \frac{w}{w+2} \], which simplifies to \[ 0 = 0 \]. Since both sides are equal, this confirms the solution is correct. It's essential to verify to rule out any potential mistakes or invalid solutions.