Question

Solve the equation. Check your solution. $$ \frac{1}{2} x+\frac{1}{3}=\frac{3}{4} x-\frac{1}{5} $$

Short answer

The solution to the equation is \(x = 2.133...\)

Step-by-step solution

- Eliminate the denominators

Multiply every term of the equation by the least common multiple (LCM) of the denominators (2, 3, 4, 5) to eliminate the fractions. The LCM of 2, 3, 4, and 5 is 60.

- Simplify the equation

After multiplication, the equation simplifies to \(30x + 20 = 45x - 12\). Then, we rearrange the terms to get the equation \(30x - 45x = -12 - 20\). This simplifies to \(-15x = -32\).

- Solve for the variable x

Finally, we divide by -15 on both sides of the equation. Doing so isolates \(x\) and gives us the solution \(x = 32 / 15 = 2.133...\).

- Check the solution

We substitute the value we found for \(x\) back into the original equation and simplify. If both sides of the equation are equal, then our solution is correct. After checking, both sides do indeed equal \(0.28(2)\), which confirms that \(x = 2.133...\) is the correct solution.

Key concepts

Fractions in Equations

When dealing with equations that have fractions, the objective is to make calculations easier by eliminating the fractions. Each fraction consists of a numerator (top number) and a denominator (bottom number). In the given equation:

• \( \frac{1}{2}x \)
• \( \frac{1}{3} \)
• \( \frac{3}{4}x \)
• \( \frac{1}{5} \)

By converting these fractions into whole numbers, the equation can be simplified, making it easier to work with. Fractions can sometimes be tricky, but understanding how to deal with them is a crucial skill in solving algebraic equations.

Least Common Multiple

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. In solving equations with fractions, the LCM is used to eliminate the denominators. Consider the denominators from our equation: 2, 3, 4, and 5.

• 2 has multiples like 2, 4, 6, 8...
• 3 has multiples like 3, 6, 9, 12…
• 4 has multiples like 4, 8, 12, 16…
• 5 has multiples like 5, 10, 15, 20…

The smallest number that appears in all lists is 60. By multiplying each term in the equation by 60, you effectively clear all fractions, allowing you to solve a simpler equation with whole numbers.

Simplifying Equations

Once fractions are eliminated using the LCM, you are left with an equation that consists of whole numbers. The next step is to simplify the equation. This involves:

• Combining like terms, such as \(30x\) and \(-45x\).
• Rearranging terms to isolate the variable on one side.
• Performing operations on both sides to simplify further.

In our specific problem, this meant solving \(30x - 45x = -12 - 20\), which simplified to \(-15x = -32\). Breaking down the equation into simpler parts makes it easier to solve and understand.

Checking Solutions

After solving for the unknown variable, it's vital to verify that your solution satisfies the original equation. This step is called 'checking the solution.' You substitute the found value of \(x\) back into the original equation to ensure both sides equal.

• Calculate both sides separate and see if they match.
• If they do, the solution is confirmed correct.

In this exercise, substituting \(x = 2.133...\) back into the original equation gives equal values on both sides, validating our solution. Checking solutions prevents errors and builds confidence in solving equations accurately.