Solving Two-Step Equations A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Two-step equations, a common type, require a systematic approach to isolate the variable and find its value. This article serves as a comprehensive guide to solving two-step equations, breaking down the process into clear, manageable steps. We'll tackle the equation $-\frac{7}{4} + \frac{1}{4}x = 2$ as a practical example, demonstrating each step involved.

Understanding the Equation Structure

Before we dive into the solution, let's analyze the equation $-\frac{7}{4} + \frac{1}{4}x = 2$. This is a two-step equation because it necessitates two operations to isolate the variable 'x'. The equation consists of several key components:

• Variable: The unknown value we aim to find, represented by 'x'.
• Coefficient: The number multiplied by the variable, in this case, $\frac{1}{4}$.
• Constant Terms: Numbers that stand alone, such as $-\frac{7}{4}$ and 2.
• Operations: The mathematical processes involved, including addition and multiplication.

The ultimate goal is to isolate 'x' on one side of the equation, revealing its value. To achieve this, we'll employ inverse operations, which undo the operations present in the equation. For instance, the inverse of addition is subtraction, and the inverse of multiplication is division. By applying these inverse operations strategically, we can systematically unravel the equation and solve for 'x'. Remember, the key is to maintain balance by performing the same operation on both sides of the equation, ensuring that the equality remains valid throughout the solution process. With a clear understanding of the equation's structure and the concept of inverse operations, we're well-equipped to tackle the steps involved in solving for 'x'. Let's move on to the first step: isolating the variable term.

Step 1: Isolate the Variable Term

Our primary objective in solving this equation is to isolate the term containing the variable, which in this case is $\frac{1}{4}x$. To achieve this, we need to eliminate the constant term that is added to or subtracted from it. In the equation $-\frac{7}{4} + \frac{1}{4}x = 2$, the constant term is $-\frac{7}{4}$.

To eliminate $-\frac{7}{4}$, we perform the inverse operation, which is adding $\frac{7}{4}$ to both sides of the equation. This maintains the balance of the equation, ensuring that both sides remain equal. The equation then transforms as follows:

$-\frac{7}{4} + \frac{1}{4}x + \frac{7}{4} = 2 + \frac{7}{4}$

On the left side, $-\frac{7}{4}$ and $+\frac{7}{4}$ cancel each other out, leaving us with just the variable term $\frac{1}{4}x$. On the right side, we need to add 2 and $\frac{7}{4}$. To do this, we first convert 2 into a fraction with a denominator of 4, which gives us $\frac{8}{4}$. Now we can add the fractions:

$\frac{8}{4} + \frac{7}{4} = \frac{15}{4}$

Therefore, our equation now simplifies to:

$\frac{1}{4}x = \frac{15}{4}$

This step has successfully isolated the variable term on one side of the equation. Now we can proceed to the next step, which involves isolating the variable itself by dealing with its coefficient. The goal is to get 'x' completely alone on one side of the equation, revealing its value. By carefully applying inverse operations, we're steadily progressing towards the solution. Understanding this step is crucial, as it lays the foundation for the subsequent steps in solving two-step equations. The ability to isolate the variable term efficiently is a key skill in algebra, enabling us to tackle more complex equations with confidence. Let's move forward and learn how to isolate the variable itself.

Step 2: Isolate the Variable

Now that we have successfully isolated the variable term, $\frac{1}{4}x = \frac{15}{4}$, our next crucial step is to isolate the variable 'x' itself. Currently, 'x' is being multiplied by the coefficient $\frac{1}{4}$. To undo this multiplication, we need to perform the inverse operation, which is division. However, instead of dividing by a fraction, it is often more convenient to multiply by the reciprocal of the fraction.

The reciprocal of $\frac{1}{4}$ is $\frac{4}{1}$, which is simply 4. So, to isolate 'x', we will multiply both sides of the equation by 4. This ensures that we maintain the balance of the equation, keeping both sides equal:

$4 \cdot \frac{1}{4}x = 4 \cdot \frac{15}{4}$

On the left side, multiplying $\frac{1}{4}x$ by 4 cancels out the fraction, leaving us with just 'x'. On the right side, we multiply $\frac{15}{4}$ by 4. This can be visualized as:

$\frac{4}{1} \cdot \frac{15}{4} = \frac{4 \cdot 15}{1 \cdot 4} = \frac{60}{4}$

Simplifying $\frac{60}{4}$, we get 15.

Therefore, our equation now simplifies to:

x = 15

We have successfully isolated the variable 'x' and found its value! This step demonstrates the power of using inverse operations to systematically solve equations. By multiplying both sides of the equation by the reciprocal of the coefficient, we effectively undid the multiplication and revealed the value of 'x'. Understanding how to isolate the variable is a fundamental skill in solving algebraic equations, and it forms the basis for tackling more complex problems. With this step completed, we have arrived at the solution. However, it is always a good practice to verify our solution by substituting it back into the original equation. Let's proceed to the verification step to ensure our answer is correct.

Step 3: Verify the Solution

After solving an equation, it's always a prudent step to verify the solution. This process involves substituting the value we found for the variable back into the original equation to ensure it holds true. In our case, we found that x = 15, and our original equation was $-\frac{7}{4} + \frac{1}{4}x = 2$.

Let's substitute x = 15 into the original equation:

$-\frac{7}{4} + \frac{1}{4}(15) = 2$

Now, we simplify the left side of the equation:

$-\frac{7}{4} + \frac{15}{4} = 2$

Combining the fractions on the left side, we get:

$\frac{-7 + 15}{4} = 2$

$\frac{8}{4} = 2$

Simplifying the fraction, we have:

$2 = 2$

As we can see, the left side of the equation equals the right side when we substitute x = 15. This confirms that our solution is correct. Verification is a crucial step in the problem-solving process, especially in mathematics. It provides us with confidence in our answer and helps us catch any potential errors we might have made along the way. By substituting the solution back into the original equation, we can ensure that the equation remains balanced and that the value we found for the variable truly satisfies the equation. This step reinforces the concept of equality in equations and highlights the importance of accuracy in mathematical calculations. With the verification step successfully completed, we can confidently conclude that x = 15 is the correct solution to the equation $-\frac{7}{4} + \frac{1}{4}x = 2$.

Summary of Steps

To recap, solving the two-step equation $-\frac{7}{4} + \frac{1}{4}x = 2$ involved the following steps:

1. Isolate the Variable Term: We added $\frac{7}{4}$ to both sides of the equation to isolate the term containing 'x'.
2. Isolate the Variable: We multiplied both sides of the equation by 4, the reciprocal of $\frac{1}{4}$, to isolate 'x'.
3. Verify the Solution: We substituted x = 15 back into the original equation to confirm that it holds true.

By following these steps methodically, we successfully solved for 'x'. This systematic approach can be applied to solve a wide range of two-step equations. Understanding the underlying principles and practicing these steps will build your confidence and proficiency in algebra. Each step plays a crucial role in unraveling the equation and arriving at the correct solution. The ability to isolate the variable term and the variable itself are fundamental skills that extend beyond two-step equations and are essential for tackling more complex algebraic problems. The verification step serves as a final check, ensuring the accuracy of the solution and reinforcing the understanding of equation balancing. As you continue your mathematical journey, remember that practice and a clear understanding of the steps involved are key to mastering equation-solving. With each equation you solve, you'll further develop your problem-solving skills and gain a deeper appreciation for the elegance and logic of mathematics.

Conclusion

In conclusion, solving two-step equations is a fundamental skill in algebra. By understanding the structure of equations and applying inverse operations systematically, we can isolate the variable and find its value. The equation $-\frac{7}{4} + \frac{1}{4}x = 2$ served as a practical example, illustrating the steps involved in solving such equations. Remember to always verify your solution to ensure accuracy. Mastering these skills will empower you to tackle more complex mathematical problems with confidence.