Problem 23 - College Algebra

Question

Exercises \(17-30\) contain equations with constants in denominators. Solve each equation. $$ \frac{3 x}{5}-x=\frac{x}{10}-\frac{5}{2} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{-25}{19}\)

Step 1: Clear Fractions

Multiplying every term by 10 to clear the fractions: \(30x - 10x = x - 25\)

Step 2: Combine like terms

Combine similar terms on each side: \(20x = x - 25\)

Step 3: Isolate 'x'

Subtract 'x' from both sides to isolate 'x' to one side of the equation: \(20x - x = - 25\)

Step 4: Simplify

Simplify to get: \(19x = -25\)

Step 5: Solve for 'x'

Solve for 'x' by dividing both sides by 19: \(x = \frac{-25}{19}\)

Key Concepts

Clearing Fractions

Clearing fractions is often the first step when working with equations involving fractions. Instead of dealing with fractions directly, we can simplify our task by eliminating them. This is often done by multiplying every term in the equation by the least common denominator (LCD) of all fractions present. This process makes each fraction a whole number, simplifying the equation considerably.

• The LCD is the smallest number that all the denominators can divide into without a remainder.
• In our example, the denominators are 5, 10, and 2. The LCD of these numbers is 10.

By multiplying each term by 10, we eliminate the fractions, making the equation easier to work with: Begin with equation: \( \frac{3x}{5} - x = \frac{x}{10} - \frac{5}{2} \)
After multiplying by 10: \( 30x - 10x = x - 25 \). The fractions have been cleared, and we're left with a simpler equation.

Combining Like Terms

Once the equation is free of fractions, the next step is to simplify further by combining like terms. Like terms are terms in an equation that contain the same variable raised to the same power. By gathering these terms together, the equation becomes less complex, making it easier to solve.

• Look for terms on the same side of the equation that can be combined.
• In our example: \(30x - 10x = x - 25\)
• The term \(30x\) and \(-10x\) are like terms because they both contain \(x\).

Combine \(30x\) and \(-10x\) together to get \(20x = x - 25\). By doing this, we've consolidated the number of terms, helping us move closer to isolating the variable.

Isolating Variables

Isolating the variable is a critical step in solving any algebraic equation as it allows us to solve for the unknown directly. This involves using basic algebraic operations to manipulate the equation so that the variable, often 'x', is on one side, and all constants are on the other.

• Subtract \(x\) from both sides: \(20x - x = -25\)
• This step leaves us with \(19x = -25\), where the variable is partly isolated.

The goal is to have \(x\) by itself, simplifying further gives us a clear path to solving for \(x\). It's important to keep equations balanced by performing identical operations on both sides.

Simplifying Equations

Simplifying an equation involves reducing it to its simplest form. This step makes it easier to identify the solution to the problem. Typically, this involves reducing coefficients and bringing the equation to a basic form where the variable can be cleanly isolated.
Our example already looks simple after isolating \(x\), and now, it needs one final touch for the solution:

• From the equation \(19x = -25\), divide both sides by 19.
• This operation reveals the value of \(x\), giving us \(x = \frac{-25}{19}\).

Remember, division in algebra helps bring the equation to a single variable equal to a number, providing a clear solution. Thus, simplifying not only solves the equation but also checks and confirms the accuracy of your solution.