Problem 22 - Algebra and Trigonometry

Question

Contain linear equations with constants in denominators. Solve each equation. $$\frac{x}{2}=\frac{3 x}{4}+5$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{x}{2}=\frac{3x}{4}+5\) is \(x = -20\).

Step 1: Clear the Fractions

Multiply every term by the common denominator, which is 4 in this case. This gives: \(2x = 3x + 20\).

Step 2: Collect the Variables on one Side

Subtract '3x' from both sides: \(-x = 20\).

Step 3: Isolate the Coefficient of 'x'

To isolate 'x', multiply both sides by -1: \(x = -20\).

Key Concepts

Clearing Fractions in Equations

When solving linear equations containing fractions, the first step is to eliminate the fractions. This simplifies the equation, making it easier to solve. The process of eliminating fractions is called 'clearing fractions'. You can clear fractions by finding a common denominator for all the terms in the equation and then multiplying each term by this common denominator.

This tactic not only simplifies the equation but also ensures that you're working with whole numbers, which reduces the risk of arithmetic errors. It’s important to apply this multiplication across the entire equation to keep it balanced. For example, if you have the equation \( \frac{x}{2} = \frac{3x}{4} + 5 \) and the common denominator is 4, multiply every term by 4 to clear the fractions. As a result, the equation becomes \( 2x = 3x + 20 \) without any denominators to worry about.

Isolating Variables

Once fractions are cleared, the next goal in solving linear equations is to isolate the variable. This means getting the variable on one side of the equal sign and all other numbers on the opposite side. To isolate variables effectively, use basic algebraic operations: addition, subtraction, multiplication, and division.

In the given example, after clearing fractions, we have \( 2x = 3x + 20 \). To isolate \( x \), we need to get all terms with \( x \) on one side and constants on the other. Therefore, we subtract \( 3x \) from both sides, yielding \( -x = 20 \). Remember to do the same operation on both sides to maintain equality. Lastly, we can multiply by -1 to solve for \( x \) resulting in \( x = -20 \), giving us the isolated variable.

Linear Equations Step by Step Solution

Solving linear equations with a step by step solution is an excellent approach for students. It breaks down the problem into smaller, more manageable steps, each building upon the last. This methodical process reduces complexity and helps students keep track of their operations.

Take our initial equation \( \frac{x}{2} = \frac{3x}{4} + 5 \). The step by step solution would include: (1) Clear the fractions. (2) Collect like terms to one side, which involves moving all variable terms to one side and constants to the other. (3) Isolate the variable by dividing or multiplying to solve for the variable. Following these ordered steps leads to a cleaner and more accurate solution.

Common Denominator in Linear Equations

The common denominator is a key concept when dealing with fractions in linear equations. It's the least common multiple of all the denominators involved in the equation. Finding a common denominator is crucial for clearing fractions because it allows us to combine and compare terms by converting them into equivalent fractions with the same denominator.

To determine the common denominator, list out the multiples of each denominator and choose the smallest number that's a multiple of all the denominators. For example, if your denominators are 2 and 4, the common denominator is 4 because it's the least common multiple. Ensuring all terms have this common denominator before proceeding with operations ensures that all the fractions involved are effectively managed.