Problem 24 - College Algebra

Question

Exercises \(17-30\) contain equations with constants in denominators. Solve each equation. $$ 2 x-\frac{2 x}{7}=\frac{x}{2}+\frac{17}{2} $$

Step-by-Step Solution

Answer

\(x = 7\)

Step 1: Identify the common multiple

Determine the least common multiple (LCM) of the denominators. Here, the denominators are 2 and 7, and their LCM is 14.

Step 2: Multiply both sides by the LCM

By multiplying every term by 14, we eliminate the denominator for easier management of the equation. This yields: \(14*(2x) -14*\(\frac{2x}{7}\) = 14*\(\frac{x}{2}\) + 14*\(\frac{17}{2}\). Simplify to achieve: \(28x -4x = 7x + 119\)

Step 3: Rearrange the equation

Formulate the equation into one side equals to zero. This yields: \(28x - 4x - 7x - 119 = 0\), which simplifies to: \(17x - 119 = 0\).

Step 4: Solve for the unknown

Add 119 to both sides: \(17x = 119\). Then, divide both sides by 17 to find the solution: \(x = \frac{119}{17}\)

Key Concepts

Understanding Least Common Multiple (LCM)

One fundamental concept in algebra is finding the least common multiple, often abbreviated as LCM. The LCM of two or more numbers is the smallest number that is a multiple of each of the numbers. To simplify algebraic equations involving fractions, LCM is used to eliminate denominators, making an equation easier to solve.

For example, if we take the numbers 2 and 7, their multiples would be:

• 2: 2, 4, 6, 8, 10, 12, 14, 16, ...
• 7: 7, 14, 21, 28, 35, ...

The smallest common multiple they share is 14. Multiplying each term in an equation by the LCM ensures that all denominators are eliminated since each term becomes a whole number. This step is crucial before progressing with solving the equation.

Eliminating Denominators for Simpler Equations

Eliminating denominators from equations streamlines the path to finding solutions. When we deal with fractions in equations, distributing the LCM to each term, as we did with 14 in our example, effectively 'clears' the fractions. The process transforms the equation into a simpler form where we can apply the usual algebraic rules without the additional complications fractions bring.

By multiplying through by the LCM, we are not changing the equation’s balance; we are merely scaling it up to a version without fractions. The term-by-term multiplication leads to a straightforward linear equation which can then be rearranged and solved using basic algebraic operations.

Simplifying Equations to Solve for Variables

After denominators are eliminated, it’s time to simplify the equation. This simplification typically involves combining like terms and arranging them such that all the terms involving the variable are on one side, and all constant terms are on the other. In our example, after eliminating fractions, we combine like terms to get an equation in the form of \(ax + b = 0\), where \(a\) and \(b\) are coefficients.

Finally, solving for the variable requires isolating it on one side of the equation. You do this by performing inverse operations, such as addition or subtraction to move the constant term, followed by division or multiplication to isolate the variable. Remember to perform each operation to both sides of the equation to maintain equality. The goal is to arrive at a simple statement, such as \(x = c\), which provides the solution to the original equation.