Problem 14 - Intermediate Algebra

Question

Solve equation. \(\frac{2}{3}+\frac{10}{a+2}=4\)

Step-by-Step Solution

Answer

The solution is \(a = 1\).

Step 1: Isolate the Fraction

Subtract \(\frac{2}{3}\) from both sides of the equation to isolate the fractional term containing the unknown variable.\[\frac{10}{a+2} = 4 - \frac{2}{3}\]

Step 2: Simplify the Right Side

Calculate the right-hand side of the equation. Convert \(4\) into a fraction with a denominator of \(3\) to facilitate subtraction.\[4 = \frac{12}{3}\]Substitute this into the equation and subtract: \[4 - \frac{2}{3} = \frac{12}{3} - \frac{2}{3} = \frac{10}{3}\]Thus, the equation becomes: \[\frac{10}{a+2} = \frac{10}{3}\]

Step 3: Clear the Fraction

Since the numerators are equal, set the denominators equal to each other to solve for \(a\).\[ a+2 = 3 \]

Step 4: Solve for the Variable

Subtract \(2\) from both sides of the equation to find \(a\).\[ a = 3 - 2 = 1 \]

Step 5: Verify the Solution

Substitute \(a = 1\) back into the original equation to check: \[\frac{2}{3} + \frac{10}{1+2} = \frac{2}{3} + \frac{10}{3} = \frac{12}{3} = 4\]The equation holds true, verifying that the solution is correct.

Key Concepts

Understanding Linear Equations

Linear equations are the foundation of many mathematical concepts. They are equations of the first degree, which means they contain only the highest power of the variable as one.
This form of equation is often simple and straightforward to solve. In this particular exercise, we have a linear equation involving a fraction. The equation given is \( \frac{2}{3}+\frac{10}{a+2}=4 \). Here, the variable \( a \) needs to be determined.
To solve it, we often start by isolating terms containing the variable. This involves various algebraic manipulations to simplify the equation.
Linear equations are particularly important because they model relationships where one quantity depends on another in a proportional manner. Understanding how to solve them is essential for tackling more complex problems in algebra.

Navigating Fractions in Equations

Fractions can often make an equation seem daunting, but they can be managed easily with some basic strategies. In our equation \( \frac{10}{a+2}=\frac{10}{3} \), fractions are used to express the ratio of quantities.
To simplify equations with fractions efficiently:

• Find a common denominator if you're tasked with adding or subtracting fractions.
• Clear fractions by multiplying every term by the least common denominator (LCD).
• Focus on isolating the fraction by getting terms on one side of the equation.

As seen in the solution steps, once the fractions are isolated, it becomes easier to proceed with other algebraic manipulations. Mastery of fractions is crucial because they frequently appear in various mathematical equations and real-life applications.

Techniques of Algebraic Manipulation

Algebraic manipulation is all about rearranging the equation to isolate the variable and find its value. This process involves several key techniques that can be systematically applied:

• Subtracting or adding terms: Move terms from one side of the equation to the other to isolate the term with the variable.
• Clearing fractions: Multiply every term by the LCD to eliminate fractions for easier calculation.
• Balancing both sides: Ensure every operation you perform is done on both sides of the equation to maintain equality.

In the exercise, after isolating and simplifying, we set the numerators equal, which further reduces the problem to a simple arithmetic operation.
As a result, algebraic manipulation stands as a critical skill, enabling one to solve equations methodically and accurately. This skill helps in understanding the structure of equations and in developing strategies to solve them.