Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution. Begin your work by rewriting each equation without fractions. \(\frac{3 x}{5}-\frac{2}{5}=\frac{x}{3}+\frac{2}{5}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3\).

Step 1: Remove Fractions

To remove fractions, multiply the entire equation by the least common multiple (LCM) of all denominators involved. Here, denominators are 5 and 3, so their LCM is 15. So we multiply the whole equation by 15, which gives: \(9x - 6 = 5x + 6\)

Step 2: Simplify and Solve

Now, simplify the equation by moving the terms involving 'x' to one side of the equation and the constant terms to the other. We do this by subtracting 5x from both sides, and also adding 6 to both sides to get \(4x = 12\). Then, to solve for 'x', divide both sides by 4 to get \(x = 3\)

Step 3: Check the Solution

Finally, confirm the solution by substituting \(x = 3\) back into the original equation. Plug it into the original equation: \(\frac{3*3}{5}-\frac{2}{5}=\frac{3}{3}+\frac{2}{5}\), which simplifies to \(1 = 1\), confirming that \(x = 3\) is the correct solution.

Key Concepts

Removing Fractions

Fractions can often make solving equations feel complicated, so it helps to eliminate them early in the process. The best way to do this is by finding a number that both denominators divide into evenly, allowing you to clear the fractions by multiplying.

• Identify the denominators in your equation. In this case, they are 5 and 3.
• Find a common number that can absorb these fractions into whole numbers. This is called the least common multiple (LCM).
• Multiply each term in the equation by this LCM, converting every fraction into a straightforward integer.

After multiplying each term by 15, the fractions vanish and the equation becomes much simpler: \(9x - 6 = 5x + 6\). No tricky fractions make life easier and solving the equation becomes more straightforward.

Least Common Multiple

Finding the least common multiple (LCM) is a crucial step when dealing with fractions in equations. It helps in eliminating the fractions, making the equation simpler to solve. Here’s how you can find it:

• List the multiples of each number involved. For example, for 5 it's 5, 10, 15, etc., and for 3 it's 3, 6, 9, 12, 15, etc.
• Find the smallest number that appears in both lists of multiples. In this case, it's 15.

By using the LCM to multiply through the equation, you clear out the fractions. This means instead of solving messy fractions, you deal with whole numbers. This change significantly eases the process of solving the equation.

Checking Solutions

No equation-solving is complete without double-checking that your solution is correct. Once you’ve solved for a variable, it's essential to substitute your found value back into the original equation to ensure it balances.
Here's how to check your solution:

• Take the value of \(x\) you found—in this case, 3—and substitute it back into the original equation: \(\frac{3 \times 3}{5} - \frac{2}{5} = \frac{3}{3} + \frac{2}{5}\).
• Simplify both sides of the original equation. If both sides turn out to be equal, your solution is correct.

In our example, substituting \(x = 3\) leads to both sides simplifying to 1, confirming that \(x = 3\) is indeed the solution. Checking solutions ensures your work is accurate and gives you the confidence that you, in fact, have the correct answer.

Simplifying Equations

Simplifying equations is all about making the problem easier to solve by reducing the complexity of the equation. After you remove fractions, it's helpful to isolate the variable on one side.
Here's the process:

• Gather all terms with the variable on one side by using addition or subtraction. This often involves moving constants to the opposite side.
• Once you've isolated terms involving the variable, simplify the equation into a basic form.
• When the equation is reduced to a simple form like \(4x = 12\), solve for \(x\) by performing division or multiplication as necessary.

The main goal is to transform complex equations into something manageable and solvable, turning the math into something easy to work with and understand. By simplifying, you focus on what really matters, the variable and its value.