Solving Linear Equations A Step-by-Step Guide To 4x/3 - (x-1)/2 = 1 1/4

Introduction to Linear Equations

In the realm of mathematics, linear equations form the bedrock of algebraic problem-solving. These equations, characterized by variables raised to the power of one, represent relationships between quantities that can be visualized as straight lines on a graph. Mastering the techniques to solve linear equations is crucial for anyone venturing into higher levels of mathematics, science, or engineering. This article delves into the step-by-step process of solving a specific linear equation: $\frac{4x}{3} - \frac{x-1}{2} = 1\frac{1}{4}$. We will explore the underlying principles, the methods employed, and the rationale behind each step, ensuring a comprehensive understanding of the solution.

This particular equation involves fractions, which might initially seem daunting. However, by applying the fundamental principles of algebra and employing a systematic approach, we can navigate through these fractions and arrive at the solution. The core strategy involves eliminating the fractions by finding a common denominator, simplifying the equation, and then isolating the variable 'x' to determine its value. Understanding these steps not only solves this specific problem but also equips you with the tools to tackle a wide array of linear equations. Moreover, the process highlights the importance of accuracy, attention to detail, and a methodical approach in mathematical problem-solving. So, let’s embark on this journey of solving this linear equation and unravel the mathematical concepts it embodies.

Step 1: Convert Mixed Number to an Improper Fraction

Before we commence solving the equation $\frac{4x}{3} - \frac{x-1}{2} = 1\frac{1}{4}$, the first crucial step involves transforming the mixed number, $1\frac{1}{4}$, into an improper fraction. This conversion is essential because it simplifies the subsequent algebraic manipulations and allows us to work with a single fractional term rather than a combination of a whole number and a fraction. The process of converting a mixed number to an improper fraction involves multiplying the whole number part by the denominator of the fractional part, adding the numerator, and then placing the result over the original denominator. In this case, we have $1\frac{1}{4}$. To convert it, we multiply the whole number 1 by the denominator 4, which gives us 4. Then, we add the numerator 1, resulting in 5. Finally, we place this sum over the original denominator 4, yielding the improper fraction $\frac{5}{4}$.

Therefore, $1\frac{1}{4}$ is equivalent to $\frac{5}{4}$. This conversion now allows us to rewrite the original equation as $\frac{4x}{3} - \frac{x-1}{2} = \frac{5}{4}$. This revised form of the equation sets the stage for the next steps in solving for 'x'. By converting the mixed number into an improper fraction, we have streamlined the equation and made it more amenable to algebraic manipulation. This seemingly simple step is a cornerstone of solving equations involving mixed numbers and underscores the importance of ensuring all terms are in a consistent format before proceeding with further calculations. The transition to improper fractions often makes the equation clearer and easier to manage, paving the way for a smoother solution process. Now that we have successfully converted the mixed number, we can move on to the next phase of the solution: eliminating the fractions by finding a common denominator.

Step 2: Find the Least Common Multiple (LCM)

With the equation now in the form $\frac{4x}{3} - \frac{x-1}{2} = \frac{5}{4}$, our next critical step is to eliminate the fractions. To achieve this, we need to find the Least Common Multiple (LCM) of the denominators, which are 3, 2, and 4. The LCM is the smallest number that is a multiple of each of these denominators. Identifying the LCM is paramount because it allows us to multiply both sides of the equation by a single value that will cancel out all the denominators, thereby transforming the equation into a simpler, fraction-free form.

To find the LCM of 3, 2, and 4, we can list the multiples of each number and identify the smallest multiple they all share. The multiples of 3 are 3, 6, 9, 12, 15, and so on. The multiples of 2 are 2, 4, 6, 8, 10, 12, and so on. The multiples of 4 are 4, 8, 12, 16, and so on. By examining these lists, we can see that the smallest number that appears in all three lists is 12. Therefore, the LCM of 3, 2, and 4 is 12. This means that 12 is the smallest number that is divisible by 3, 2, and 4. Now that we have determined the LCM, we will use it to multiply both sides of the equation. This process will effectively eliminate the fractions, making the equation easier to solve. Finding the LCM is a fundamental skill in algebra, particularly when dealing with equations involving fractions. It allows us to transform complex equations into simpler ones, making them more manageable and less prone to errors. In the next step, we will multiply both sides of the equation by 12, effectively clearing the fractions and setting the stage for further simplification and the isolation of the variable 'x'.

Step 3: Multiply Both Sides of the Equation by the LCM

Having determined that the Least Common Multiple (LCM) of the denominators 3, 2, and 4 is 12, the next crucial step is to multiply both sides of the equation $\frac{4x}{3} - \frac{x-1}{2} = \frac{5}{4}$ by this LCM. This operation is pivotal because it eliminates the fractions, transforming the equation into a more manageable form that is easier to solve. When we multiply both sides of the equation by 12, we are essentially applying the multiplication property of equality, which states that if we perform the same operation on both sides of an equation, the equality remains valid.

Multiplying the left side of the equation, $\frac{4x}{3} - \frac{x-1}{2}$, by 12 involves distributing the 12 to each term. This gives us $12 * \frac{4x}{3} - 12 * \frac{x-1}{2}$. Simplifying each term, we have: $(12/3) * 4x - (12/2) * (x-1)$, which simplifies further to $4 * 4x - 6 * (x-1)$, and finally to $16x - 6(x-1)$.

On the right side of the equation, we multiply $\frac{5}{4}$ by 12, which gives us $12 * \frac{5}{4}$. Simplifying this, we have $(12/4) * 5$, which equals $3 * 5$, and thus simplifies to 15.

Therefore, after multiplying both sides of the equation by the LCM, 12, the equation transforms from $\frac{4x}{3} - \frac{x-1}{2} = \frac{5}{4}$ to $16x - 6(x-1) = 15$. This transformation is a significant milestone in solving the equation. By eliminating the fractions, we have simplified the equation and made it more amenable to further algebraic manipulations. The next step will involve distributing the -6 across the parentheses on the left side of the equation, which will allow us to combine like terms and continue the process of isolating the variable 'x'. This step of multiplying by the LCM is a fundamental technique in solving equations involving fractions and demonstrates the power of algebraic manipulation in simplifying complex problems.

Step 4: Distribute and Simplify

Following the elimination of fractions by multiplying both sides of the equation by the Least Common Multiple (LCM), we now have the equation in the form $16x - 6(x-1) = 15$. The next critical step is to distribute the -6 across the parentheses on the left side of the equation. This distribution is a direct application of the distributive property of multiplication over subtraction, which states that $a(b - c) = ab - ac$. Applying this property is essential to remove the parentheses and combine like terms, bringing us closer to isolating the variable 'x'.

Distributing the -6 in the term -6(x-1) involves multiplying -6 by both x and -1. This yields -6 * x and -6 * (-1). Thus, -6(x-1) expands to -6x + 6. Substituting this expansion back into the equation, we have $16x - 6x + 6 = 15$. Now that the parentheses are removed, we can proceed to simplify the equation by combining like terms.

On the left side of the equation, we have two terms involving 'x': 16x and -6x. Combining these terms involves adding their coefficients, which are 16 and -6 respectively. Therefore, 16x - 6x simplifies to 10x. The equation now becomes $10x + 6 = 15$. This simplified form is much easier to work with, and we are now one step closer to isolating 'x'. The process of distributing and simplifying is a fundamental algebraic technique that is used extensively in solving equations. It allows us to transform complex expressions into simpler forms, making it easier to identify and manipulate the variables. In the next step, we will isolate the term involving 'x' by subtracting 6 from both sides of the equation. This will further simplify the equation and bring us closer to the final solution.

Step 5: Isolate the Variable Term

Having simplified the equation to $10x + 6 = 15$, the next key step in solving for 'x' is to isolate the variable term, which in this case is 10x. To achieve this, we need to eliminate the constant term, +6, from the left side of the equation. This is accomplished by applying the subtraction property of equality, which dictates that we can subtract the same value from both sides of the equation without altering the equality. In this instance, we will subtract 6 from both sides of the equation.

Subtracting 6 from the left side of the equation, $10x + 6$, we get $10x + 6 - 6$, which simplifies to 10x. This effectively isolates the variable term on the left side. On the right side of the equation, we subtract 6 from 15, resulting in $15 - 6$, which equals 9.

Therefore, after subtracting 6 from both sides of the equation, we are left with $10x = 9$. This simplified equation represents a significant milestone in our solution process. The variable term, 10x, is now isolated on one side of the equation, making it easier to solve for 'x'. We are now one step away from determining the value of 'x'. The process of isolating the variable term is a fundamental technique in solving algebraic equations. It involves strategically applying inverse operations to eliminate constants and coefficients from the side of the equation containing the variable. In the next step, we will divide both sides of the equation by the coefficient of 'x' to finally solve for the value of 'x'. This will complete the process of isolating the variable and provide us with the solution to the equation.

Step 6: Solve for x

With the equation now simplified to $10x = 9$, the final step in solving for 'x' involves isolating 'x' completely. This is achieved by dividing both sides of the equation by the coefficient of 'x', which in this case is 10. This operation is based on the division property of equality, which states that dividing both sides of an equation by the same non-zero number maintains the equality.

Dividing the left side of the equation, 10x, by 10 results in $\frac{10x}{10}$, which simplifies to x. This isolates the variable 'x' on the left side. On the right side of the equation, we divide 9 by 10, resulting in $\frac{9}{10}$. This fraction represents the value of 'x'.

Therefore, after dividing both sides of the equation by 10, we find that $x = \frac{9}{10}$. This is the solution to the original equation, $\frac{4x}{3} - \frac{x-1}{2} = 1\frac{1}{4}$. We have successfully navigated through the steps of converting the mixed number to an improper fraction, finding the Least Common Multiple, multiplying both sides by the LCM, distributing and simplifying, isolating the variable term, and finally, solving for 'x'.

The value of x, $\frac{9}{10}$, can also be expressed as a decimal, which is 0.9. This decimal form provides an alternative representation of the solution. The process of solving for 'x' demonstrates the power of algebraic manipulation in isolating variables and finding solutions to equations. This final step completes the solution process, providing us with the value of 'x' that satisfies the original equation. The ability to solve for variables is a fundamental skill in algebra and is essential for tackling more complex mathematical problems.

Conclusion

In summary, solving the linear equation $\frac{4x}{3} - \frac{x-1}{2} = 1\frac{1}{4}$ involved a series of carefully orchestrated steps, each building upon the previous one to ultimately isolate the variable 'x' and determine its value. The journey began with converting the mixed number, $1\frac{1}{4}$, into an improper fraction, $\frac{5}{4}$, which streamlined the equation for further manipulation. We then identified the Least Common Multiple (LCM) of the denominators, which was 12, and multiplied both sides of the equation by this LCM to eliminate the fractions. This crucial step transformed the equation into a more manageable form, $16x - 6(x-1) = 15$.

Following the elimination of fractions, we distributed the -6 across the parentheses, resulting in $16x - 6x + 6 = 15$. This distribution was a key application of the distributive property, allowing us to remove the parentheses and combine like terms. Simplifying the equation further, we combined the 'x' terms, resulting in $10x + 6 = 15$. The next step involved isolating the variable term, 10x, by subtracting 6 from both sides of the equation, yielding $10x = 9$.

Finally, to solve for 'x', we divided both sides of the equation by 10, the coefficient of 'x', which gave us the solution $x = \frac{9}{10}$. This solution can also be expressed as the decimal 0.9. The entire process underscores the importance of a systematic and methodical approach to solving algebraic equations. Each step, from converting mixed numbers to isolating the variable, requires careful attention to detail and a solid understanding of algebraic principles.

This exercise not only provides the solution to this specific equation but also reinforces the fundamental techniques of solving linear equations. These techniques are applicable to a wide range of mathematical problems and are essential for anyone pursuing studies in mathematics, science, or engineering. The ability to confidently solve linear equations is a cornerstone of mathematical literacy and empowers individuals to tackle more complex challenges in the future. The step-by-step approach demonstrated here serves as a valuable framework for approaching similar problems, emphasizing the importance of breaking down complex tasks into smaller, more manageable steps. Thus, mastering these techniques is an investment in one's mathematical proficiency and problem-solving abilities.