Problem 8 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(\frac{3}{4} n=-15\)

Step-by-Step Solution

Answer

The solution to the equation is \( n = -20 \).

Step 1: Understand the Equation

The equation given is \( \frac{3}{4} n = -15 \). Here, \( \frac{3}{4} \) is multiplied by \( n \) and equals \( -15 \). Our goal is to find the value of \( n \).

Step 2: Isolate the Variable

To find \( n \), we need to isolate it by getting rid of the fraction. We can do this by multiplying both sides of the equation by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \).

Step 3: Multiply to Simplify

Multiply both sides by \( \frac{4}{3} \): \[ \left( \frac{4}{3} \right) \times \left( \frac{3}{4} n \right) = \left( \frac{4}{3} \right) \times (-15) \] On the left side, \( \frac{4}{3} \times \frac{3}{4} \) will cancel out, leaving \( n \). On the right side, calculate \( \frac{4}{3} \times (-15) \).

Step 4: Solve the Right Side

Calculate the right side: \( \frac{4}{3} \times (-15) = -20 \)Therefore, \( n = -20 \).

Step 5: Verify the Solution

Substitute \( n = -20 \) back into the original equation to verify:\( \frac{3}{4} \times (-20) = -15 \).Calculate it: \( -15 = -15 \), which is true. So, the solution is correct.

Key Concepts

Fraction Multiplication

When dealing with equations that include fractions, understanding how to multiply fractions is key. Fraction multiplication is incredibly straightforward once you know the steps.

• First, multiply the numerators (the top numbers) of the fractions to get a new numerator.
• Next, multiply the denominators (the bottom numbers) to get a new denominator.

That's it! One important thing to notice is that the multiplication process does not require a common denominator, unlike addition or subtraction. This makes multiplying fractions relatively simple.
For example, consider multiplying \( \frac{3}{4} \) with any other fraction \( \frac{a}{b} \). You would just multiply the numbers across like so: \[\frac{3 \times a}{4 \times b} = \frac{3a}{4b}. \]In the exercise, we dealt with multiplying \( \frac{4}{3} \) and \( -15 \), which does not involve fractions on both sides but handles the negative sign just like multiplying integers.

Reciprocal

The reciprocal of a number or a fraction is a value that, when multiplied with the original number or fraction, gives a result of 1. For fractions, the reciprocal is achieved by swapping the numerator and the denominator.
For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). This process is crucial when trying to isolate a variable in a fraction equation. Multiplying by the reciprocal clears the fraction, leaving the variable alone.

• Reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
• Multiplying any number by its reciprocal results in 1 (e.g., \( \frac{3}{4} \times \frac{4}{3} = 1 \)).

In this exercise, we isolated \( n \) by multiplying both sides of the equation by the reciprocal \( \frac{4}{3} \) to eliminate the fraction \( \frac{3}{4} \). This step is essential to solve for the variable in a straightforward manner.

Verifying Solutions

Once you have found a solution to an equation, especially one that includes fractions, it is important to verify that your solution is correct. Verification involves substituting your solution back into the original equation to check consistency. Here’s how you verify:

• Replace the variable in the original equation with the solution you found.
• Simplify the equation to make sure both sides equal the same value.

In our exercise, after finding that \( n = -20 \), we substituted it back:\[\frac{3}{4} \times (-20) = -15.\]Calculating the right side confirms \( -15 = -15 \), showing that the solution is indeed correct. Verification acts as a double-check to ensure no mistakes were made during the solving process.

Variables

Variables are symbols, usually letters, that represent unknown values or quantities in mathematical equations. They are placeholders for numbers we need to find.
In equations, the goal is often to solve for the variable, isolating it to find the unknown number it represents. Variables can stand alone, be multiplied by other numbers, or be part of more complex expressions.

• Common variable letters include \( n, x, y, \) etc.
• Variables can appear in equations, expressions, and inequalities.

In the given exercise, \( n \) was the variable that needed to be solved. By using techniques such as reciprocal multiplication, we transformed the equation to find the value of \( n \) that satisfies the equation, highlighting how crucial understanding variables and their manipulation is in problem-solving.