Solving The Equation (3/2)x + (9/4)x + (3/20) = 12/5 A Step-by-Step Guide

Introduction

In this comprehensive guide, we will tackle the equation $\frac{3}{2}x + \frac{9}{4}x + \frac{3}{20} = \frac{12}{5}$ and walk you through each step to arrive at the solution. This equation involves fractions and a single variable, $x$. Our goal is to isolate $x$ on one side of the equation to find its value. The solution will be expressed as an integer, a simplified fraction, or a decimal rounded to two decimal places. Understanding how to solve such equations is a fundamental skill in algebra and is crucial for various applications in mathematics and other fields. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide a clear and detailed explanation.

Step 1: Combine Like Terms

The first key step in solving the equation $\frac{3}{2}x + \frac{9}{4}x + \frac{3}{20} = \frac{12}{5}$ is to combine the like terms on the left side. Like terms are those that contain the same variable raised to the same power. In this equation, $\frac{3}{2}x$ and $\frac{9}{4}x$ are like terms because they both contain $x$ to the power of 1. To combine these terms, we need to find a common denominator. The denominators are 2 and 4, and the least common multiple (LCM) of 2 and 4 is 4. Therefore, we will convert $\frac{3}{2}x$ to an equivalent fraction with a denominator of 4. We multiply both the numerator and the denominator of $\frac{3}{2}x$ by 2, which gives us $\frac{6}{4}x$. Now we can add the two terms:

$$\frac{3}{2}x + \frac{9}{4}x = \frac{6}{4}x + \frac{9}{4}x$$

Adding the numerators while keeping the common denominator, we get:

$$\frac{6}{4}x + \frac{9}{4}x = \frac{6+9}{4}x = \frac{15}{4}x$$

So, the equation now becomes:

$$\frac{15}{4}x + \frac{3}{20} = \frac{12}{5}$$

Combining like terms simplifies the equation and makes it easier to work with. This step is a critical foundation for the subsequent steps in solving for $x$. By correctly identifying and combining like terms, we reduce the complexity of the equation and pave the way for isolating the variable.

Step 2: Isolate the Term with x

Now that we've combined the like terms, our equation looks like this: $\frac{15}{4}x + \frac{3}{20} = \frac{12}{5}$. The next essential step is to isolate the term containing $x$, which is $\frac{15}{4}x$. To do this, we need to eliminate the constant term, $\frac{3}{20}$, from the left side of the equation. We can achieve this by subtracting $\frac{3}{20}$ from both sides of the equation. This maintains the equality, as we are performing the same operation on both sides. So, we have:

$$\frac{15}{4}x + \frac{3}{20} - \frac{3}{20} = \frac{12}{5} - \frac{3}{20}$$

The $\frac{3}{20}$ on the left side cancels out, leaving us with:

$$\frac{15}{4}x = \frac{12}{5} - \frac{3}{20}$$

Now, we need to subtract the fractions on the right side. To do this, we need a common denominator. The least common multiple (LCM) of 5 and 20 is 20. So, we convert $\frac{12}{5}$ to an equivalent fraction with a denominator of 20. We multiply both the numerator and the denominator of $\frac{12}{5}$ by 4, which gives us $\frac{48}{20}$. Now we can subtract the fractions:

$$\frac{12}{5} - \frac{3}{20} = \frac{48}{20} - \frac{3}{20}$$

Subtracting the numerators while keeping the common denominator, we get:

$$\frac{48}{20} - \frac{3}{20} = \frac{48-3}{20} = \frac{45}{20}$$

We can simplify the fraction $\frac{45}{20}$ by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 5. This gives us:

$$\frac{45}{20} = \frac{45 ÷ 5}{20 ÷ 5} = \frac{9}{4}$$

So, our equation now looks like this:

$$\frac{15}{4}x = \frac{9}{4}$$

By isolating the term with $x$, we are one step closer to finding the value of $x$. This step involves careful manipulation of the equation to maintain balance and ensure we are progressing towards the correct solution.

Step 3: Solve for x

Having isolated the term with $x$, our equation now stands as $\frac{15}{4}x = \frac{9}{4}$. The final step in solving for $x$ is to eliminate the coefficient $\frac{15}{4}$ from the left side. To do this, we need to perform the inverse operation of multiplication, which is division. However, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{15}{4}$ is $\frac{4}{15}$. Therefore, we will multiply both sides of the equation by $\frac{4}{15}$. This maintains the equality of the equation:

$$\frac{4}{15} × \frac{15}{4}x = \frac{4}{15} × \frac{9}{4}$$

On the left side, $\frac{4}{15}$ and $\frac{15}{4}$ cancel each other out, leaving us with just $x$:

$$x = \frac{4}{15} × \frac{9}{4}$$

Now, we multiply the fractions on the right side. We can simplify this multiplication by canceling out common factors before multiplying. Notice that both the numerators have a factor of 4, which can be canceled out:

$$x = \frac{\cancel{4}}{15} × \frac{9}{\cancel{4}} = \frac{1}{15} × 9$$

Now we have:

$$x = \frac{9}{15}$$

We can simplify this fraction further by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

$$x = \frac{9 ÷ 3}{15 ÷ 3} = \frac{3}{5}$$

So, the solution for $x$ is $\frac{3}{5}$. However, the question asks for the answer as an integer, a simplified fraction, or a decimal rounded to two decimal places. We already have the simplified fraction, but we can also convert it to a decimal by dividing 3 by 5:

$$x = \frac{3}{5} = 0.6$$

Since 0.6 has only one decimal place, we can write it as 0.60 to express it to two decimal places.

Therefore, the solution to the equation $\frac{3}{2}x + \frac{9}{4}x + \frac{3}{20} = \frac{12}{5}$ is $x = \frac{3}{5}$ or $x = 0.60$.

Conclusion

In this guide, we have successfully solved the equation $\frac{3}{2}x + \frac{9}{4}x + \frac{3}{20} = \frac{12}{5}$ by following a step-by-step approach. We began by combining like terms, which involved finding a common denominator and adding the fractions. Next, we isolated the term containing $x$ by subtracting the constant term from both sides of the equation and simplifying the resulting fractions. Finally, we solved for $x$ by multiplying both sides of the equation by the reciprocal of the coefficient of $x$, leading us to the solution $x = \frac{3}{5}$ or $x = 0.60$. This process demonstrates the importance of understanding fundamental algebraic principles such as combining like terms, isolating variables, and performing inverse operations. These skills are essential for tackling more complex equations and problems in mathematics and various real-world applications. By mastering these techniques, you can confidently approach and solve a wide range of algebraic equations.