Problem 95 - Introductory and Intermediate Algebra for College Students 4th

Question

Suppose that you solve \(\frac{x}{5}-\frac{x}{2}=1\) by multiplying both sides by \(20,\) rather than the least common denominator, \(10 .\) Describe what happens. If you get the correct solution, why do you think we clear the equation of fractions by multiplying by the least common denominator?

Step-by-Step Solution

Answer

The solution to the equation is \(x=-\frac{10}{3}\). Multiplying by 20 also leads to the correct answer, but using the least common denominator is preferred to reduce computations and chances of mistakes.

Step 1: Identify the Equation

The given equation is \(\frac{x}{5}-\frac{x}{2}=1\). It needs to be solved for x.

Step 2: Multiply by 20

Multiply each term of the equation by 20 to get rid of the denominator. The equation becomes \(4x-10x=20\).

Step 3: Simplify and Solve for x

Solve the equation obtained in Step 2, which simplifies to -6x=20. Then, divide both sides by -6 to obtain \(x=-\frac{20}{6}\), which simplifies to \(x=-\frac{10}{3}\).

Step 4: Understanding the result

Solving by multiplying with 20 leads to a correct answer, but generally solving with the least common denominator simplifies the process and reduces chances of errors. In this case, using the multiples of the denominators as factors (5 and 2), the least common denominator is 10, which simplifies the fractions quickly and reduces calculation errors.

Key Concepts

Least Common Denominator

When dealing with fractions in equations, finding the Least Common Denominator (LCD) is often the first crucial step. The LCD is the smallest number that each denominator in the equation can divide evenly into. It's helpful because it allows you to clear the fractions from the equation altogether by multiplying every term by this number.

Taking the original problem

• \(\frac{x}{5}-\frac{x}{2}=1\)
• The denominators here are 5 and 2. The smallest number both 5 and 2 divide into without a remainder is 10.

Multiplying each term by this LCD simplifies the equation to just whole numbers. This simplifies later calculations and makes the equation easier to solve by eliminating fractions altogether.

By using the LCD, you reduce the risk of mathematical errors and the algebra becomes straightforward, often leading to a quicker and clearer solution.

Fractions

Fractions can make algebra feel intimidating, but they are just another way of representing numbers. In an algebraic equation, fractions often appear when we deal with divisions. They tell us about a part of a whole, and in equations, they're important for expressing proportions.

To solve equations involving fractions, usually one wants to clear them out to make the arithmetic simpler. This involves

• Identifying the denominators involved
• Finding a common ground, like the least common denominator
• Then multiplying each term by the chosen multiplier.

This process turns fractional coefficients into whole numbers, making the equation simpler to manipulate. By focusing on changing the form of the fractions, rather than their value, you maintain the integrity of the equation while making it easier to solve.

Equation Simplification

When simplifying equations, the goal is to reduce the equation to its simplest and most solvable form. It's about clearing away complexity, typically by eliminating fractions, combining like terms, and isolating the variable you're solving for.

In the original problem, by multiplying each term by 20, you effectively removed the fractions:

• Converting \(\frac{x}{5}-\frac{x}{2}=1\) to \(4x-10x=20\)
• This simplification left you with a linear equation that's easier to solve, yielding \(-6x=20\).

From there, solving for \(x\) becomes a matter of standard algebra:

• Isolating the variable, and
• Performing arithmetic operations like addition, subtraction, multiplication, or division.

Simplifying an equation correctly ensures that you get to the correct solution more efficiently. Remember, the key is respecting the equalities and performing operations carefully to maintain balance in the equation.