Solving Two-Step Equations 5.1 = -3x - 4.2 Step By Step Guide

In this interactive exercise, we'll walk through the process of solving a two-step equation by filling in the missing steps. We'll be focusing on the equation $5.1 = -3x - 4.2$. By strategically dragging tiles, you'll be able to follow the logical progression and arrive at the solution. This method not only provides the answer but also reinforces the fundamental algebraic principles involved in solving equations.

Understanding Two-Step Equations

Before we dive into the specifics of this equation, let's briefly discuss what two-step equations are. Two-step equations are algebraic equations that require two operations to isolate the variable. These operations typically involve addition or subtraction, followed by multiplication or division, or vice versa. The goal is always to get the variable (in this case, 'x') by itself on one side of the equation.

To effectively solve two-step equations, it’s crucial to understand the order of operations and the concept of inverse operations. The order of operations (often remembered by the acronym PEMDAS/BODMAS) dictates the sequence in which mathematical operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). However, when solving equations, we essentially reverse this order to isolate the variable.

Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. To isolate the variable, we use inverse operations to eliminate terms on the same side of the equation as the variable. This process keeps the equation balanced, ensuring that the value of the variable remains consistent.

For the equation $5.1 = -3x - 4.2$, we can see that the variable 'x' is being multiplied by -3 and then 4.2 is being subtracted. Therefore, to solve for 'x', we will need to reverse these operations. First, we'll address the subtraction by adding, and then we'll deal with the multiplication by dividing. This step-by-step approach is fundamental to solving not just two-step equations, but more complex algebraic problems as well. By understanding these core concepts, you'll be well-equipped to tackle a wide range of equations and mathematical challenges.

Step 1: Isolating the Term with the Variable

The first crucial step in solving the equation $5.1 = -3x - 4.2$ is to isolate the term that contains the variable – in this case, $-3x$. This means we need to eliminate the constant term, which is -4.2, from the right side of the equation. To do this, we employ the concept of inverse operations.

The inverse operation of subtraction is addition. Therefore, to eliminate -4.2, we need to add 4.2 to both sides of the equation. It's essential to perform the same operation on both sides to maintain the equality. This principle is a cornerstone of algebraic manipulation – whatever you do to one side of the equation, you must do to the other to keep it balanced. Imagine the equation as a weighing scale; if you add weight to one side, you must add the same weight to the other to maintain equilibrium.

So, we add 4.2 to both sides of the equation:

$5. 1 + 4.2 = -3x - 4.2 + 4.2$

This step effectively cancels out the -4.2 on the right side, leaving us with just the term containing the variable. On the left side, we perform the addition: $5.1 + 4.2 = 9.3$. Therefore, the equation now becomes:

$9. 3 = -3x$

This simplified equation is a significant milestone in solving for 'x'. We've successfully isolated the term with the variable, which means we're one step closer to finding the value of 'x'. This step highlights the importance of using inverse operations to undo the operations performed on the variable. By adding 4.2 to both sides, we strategically eliminated a term, paving the way for the next step in the solution process.

Step 2: Solving for the Variable

After successfully isolating the term with the variable in the previous step, we are now at the critical point of solving for the variable itself. Our equation stands as $9.3 = -3x$. The variable 'x' is currently being multiplied by -3. To isolate 'x', we need to perform the inverse operation of multiplication, which is division.

To eliminate the -3 from the right side of the equation, we must divide both sides by -3. Again, maintaining the balance of the equation is paramount. Dividing both sides by the same non-zero number ensures that the equality holds true. This principle of maintaining balance is a recurring theme in algebra and is crucial for accurate equation solving.

Performing the division, we get:

rac{9.3}{-3} = rac{-3x}{-3}

On the right side, the -3 in the numerator and the -3 in the denominator cancel each other out, leaving us with just 'x'. On the left side, we perform the division: $9.3 / -3 = -3.1$. Therefore, the equation simplifies to:

$-3. 1 = x$

This result signifies that we have successfully solved for 'x'. The value of 'x' that satisfies the original equation $5.1 = -3x - 4.2$ is -3.1. This step underscores the power of inverse operations in isolating variables. By dividing both sides by -3, we effectively undid the multiplication that was being performed on 'x', revealing its value. The solution -3.1 can now be substituted back into the original equation to verify its correctness.

Step 3: Verifying the Solution

Once we've found a solution to an equation, it's always a good practice to verify the solution. This step ensures that we haven't made any errors in our calculations and that the value we've obtained for the variable truly satisfies the original equation. Verification provides confidence in the accuracy of our solution and reinforces the understanding of equation solving.

To verify our solution, we substitute the value we found for 'x', which is -3.1, back into the original equation: $5.1 = -3x - 4.2$. Replacing 'x' with -3.1, we get:

$5. 1 = -3(-3.1) - 4.2$

Now, we perform the operations on the right side of the equation. First, we multiply -3 by -3.1. A negative number multiplied by a negative number results in a positive number, so we have:

$5. 1 = 9.3 - 4.2$

Next, we subtract 4.2 from 9.3:

$5. 1 = 5.1$

The left side of the equation (5.1) is equal to the right side of the equation (5.1). This equality confirms that our solution, x = -3.1, is indeed correct. The verification process demonstrates that when we substitute -3.1 for 'x' in the original equation, both sides of the equation balance out.

This step is not just a formality; it's a crucial part of the problem-solving process. By verifying our solution, we gain assurance that we've correctly applied the principles of algebra and that our answer is accurate. It also highlights the interconnectedness of the steps involved in solving equations – each step builds upon the previous one, and the final solution must align with the initial equation.

Conclusion

In this exercise, we've methodically walked through the steps to solve the two-step equation $5.1 = -3x - 4.2$. We began by understanding the concept of two-step equations and the importance of inverse operations. Then, we isolated the term with the variable by adding 4.2 to both sides of the equation. Next, we solved for 'x' by dividing both sides by -3, arriving at the solution $x = -3.1$. Finally, we verified our solution by substituting it back into the original equation, confirming its accuracy.

This step-by-step approach is a valuable strategy for tackling a wide range of algebraic equations. By breaking down the problem into manageable steps and applying the principles of inverse operations and maintaining balance, you can confidently solve for unknown variables. Remember, the key is to isolate the variable by systematically undoing the operations performed on it.

Solving equations is a fundamental skill in mathematics and has applications in various fields, from science and engineering to finance and economics. The ability to manipulate equations and solve for unknowns is a powerful tool for problem-solving and critical thinking. By mastering the techniques discussed in this exercise, you'll be well-prepared to tackle more complex mathematical challenges and apply these skills in real-world situations. The process of filling in the missing steps reinforces not only the mechanics of equation solving but also the underlying logic and reasoning. This understanding is essential for building a strong foundation in algebra and beyond.