Problem 8 - Algebra 1: Concepts and Skills

Question

Solve the equation using the cross product property. Remember to check your solutions. \(\frac{3}{u+2}=\frac{1}{u-2}\)

Step-by-Step Solution

Answer

The solution to the given equation is \(u = 4\).

Step 1: Applying Cross-multiplication

Start by cross-multiplying, which is the equivalent of multiplying the denominators of each fraction against the numerators of the other fractions. This gives the equation \(3(u - 2) = 1(u + 2)\)

Step 2: Simplifying the Equation

Expand both sides of the equation, leading to the equation \(3u - 6 = u + 2\)

Step 3: Further Simplification

Rearrange the equation in order to isolate the variable on one side. By subtracting \(u\) from both sides, and adding \(6\) to both sides, it simplifies to \(2u = 8\)

Step 4: Find the Value of the Variable

Finally, solve for \(u\) by dividing each side by \(2\), resulting in \(u = 4\)

Step 5: Check the Solution

To ensure that this solution doesn't make the denominator of the original equation equal to zero, substitute \(4\) back into the original equation. This yields \(\frac{3}{4 + 2}\) = \(\frac{1}{4 - 2}\), which simplifies to \(\frac{1}{2}\) = \(\frac{1}{2}\). The left side of the equation is equal to the right side, verifying that the solution \(u = 4\) is correct.

Key Concepts

Solving Equations

When faced with an equation, your goal is to find the value of the variable that makes the equation true. Let's dive into solving equations with a technique called cross-multiplication. This might sound complex, but it's all about clearing the fractions and simplifying.

• Start by identifying the fractions. For example, in our exercise, we have \( \frac{3}{u+2} = \frac{1}{u-2} \).
• Use cross-multiplication by multiplying the numerator of one fraction by the denominator of the other and vice versa to eliminate the fractions. This gives us the equation \( 3(u - 2) = 1(u + 2) \).
• Once cross-multiplication is done, expand and simplify the equation to isolate the variable on one side.

It’s as simple as treating the fractions like any other terms by multiplying out the denominators. This step helps you in getting a clear path to isolate and solve for the variable.

Rational Equations

Rational equations involve fractions which can make them tricky at first glance. A rational equation is just an equation that contains one or more rational expressions. Rational expressions are fractions where the numerator and the denominator are polynomials.

• The core idea with rational equations is to find a common way to eliminate the fractions, often by cross-multiplying or finding a common denominator.
• For example, in the equation \( \frac{3}{u+2} = \frac{1}{u-2} \), by cross-multiplying we convert the equation into \( 3(u - 2) = 1(u + 2) \), which is a straightforward linear equation.

Such manipulation turns the tricky fractions into easier-to-handle equations. Remember, rational equations often have restrictions related to values that make denominators zero, so watch for those hazards. It's crucial to consider these as potential traps when verifying your solutions.

Verification of Solutions

Once you have a potential solution, verify it to ensure it's correct and valid. Verification involves plugging the solution back into the original equation to see if it satisfies all parts of the equation.

• Replace the variable with the solution you found. For example, if \( u = 4 \) is our solution, substitute it back into the original equation: \( \frac{3}{4+2} = \frac{1}{4-2} \).
• Simplify both sides of the equation. In our example, both sides simplify to \( \frac{1}{2} \), proving that our solution is correct.
• Always check for undefined expressions. For rational equations, confirm that your solution doesn't make any denominator zero.

By verifying solutions, not only do you confirm accuracy, but you also gain confidence in your problem-solving process. This step is essential for avoiding mistakes and ensuring that the solution is both correct and meaningful.