Problem 31 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. Begin your work by rewriting each equation without fractions. $$\frac{x}{5}-4=-6$$

Step-by-Step Solution

Answer

\(-10\) is the solution to the given equation.

Step 1: Remove Fractions

The first thing is to eliminate the fraction from the equation. To do this, multiply every term in the equation by 5 to get rid of the denominator of the fraction. Hence, \(x - 20 = -30\).

Step 2: Solve for x

We have already eliminated the fraction. The next step is to isolate \(x\). To do this, add 20 to both sides of the equation to isolate \(x\). This gives us \(x = -10\).

Step 3: Check Your Solution

The final step is to validate our solution by substituting \(x = -10\) back into the original equation. Substituting \(x = -10\) into \(\frac{x}{5}-4=-6\), we have \(-2 - 4 = -6\), which is a true statement.

Key Concepts

Fraction Elimination

When faced with linear equations that involve fractions, a common strategy is 'fraction elimination.' In simple terms, this means getting rid of the fractions to make the equation easier to work with. Elimination is often achieved by multiplying every term in the equation by the least common denominator (LCD) of all the fractions present.

In the example, \(\frac{x}{5}-4=-6\), the LCD is 5 since it's the only denominator. Multiplying through by 5 clears the fraction, resulting in a simpler equation, \(x - 20 = -30\). This is a critical step as it reduces the chances of errors in later operations, and it can also make the process of solving the equation more intuitive for many students.

Algebraic Operations

Algebraic operations are the foundation of solving equations. These include addition, subtraction, multiplication, and division, which are used to isolate the variable and find its value. In the context of the given exercise, after eliminating the fraction, we are left with the equation \(x - 20 = -30\). To isolate \(x\), we simply undo what has been done to it by reversing the algebraic operations already applied.

So, we add 20 to both sides of \(x - 20 = -30\) to cancel the -20 that's subtracting from \(x\). This gives us \(x = -10\), which is the solution. This simple move—adding the same number to both sides—is a perfect example of how we use algebraic operations to keep the equation balanced while we solve.

Solution Checking

Verifying our answers is just as important as solving them. 'Solution checking' involves substituting the value we found back into the original equation to make sure it holds true.

For the problem \(\frac{x}{5}-4=-6\), after we've calculated \(x = -10\), checking our solution means replacing \(x\) with -10 in the original equation. If both sides of the equation equal after this substitution, it confirms that our solution is correct. In this case, \(-2 - 4 = -6\) checks out, assuring us that \(x = -10\) is indeed the right answer.

Effective solution checking is not just a great habit for confirming answers, it also reinforces understanding of the relationships within the equation.