Solve For X: 1.8 - 3.7x = -4.2x + 0.3
Hey math enthusiasts! Ever stared at an equation and wondered how to crack it? Today, we're diving deep into solving for an unknown variable, specifically 'x', in the equation 1.8 - 3.7x = -4.2x + 0.3. This might look a little daunting with all those decimals, but don't worry, we'll break it down step-by-step. Our goal is to isolate 'x' on one side of the equation, revealing its true value. Think of it like a puzzle; each step brings us closer to the solution. We'll use fundamental algebraic principles to move terms around, combine like terms, and eventually arrive at our answer. Ready to flex those brain muscles and find the value of 'x'? Let's get started on this mathematical journey!
Understanding the Equation and Our Goal
The equation we're working with is 1.8 - 3.7x = -4.2x + 0.3. Our primary mission is to find the specific numerical value for 'x' that makes this equation true. In simpler terms, we want to find out what number 'x' has to be so that when you plug it into both the left side (1.8 - 3.7x) and the right side (-4.2x + 0.3), you get the exact same result. This is the essence of solving algebraic equations. We're not just manipulating symbols; we're uncovering a hidden truth within the numbers. The presence of decimals might make you pause, but the process remains the same. We'll use the rules of algebra, which are like the universal laws of numbers, to guide us. These rules ensure that whatever operation we perform on one side of the equation, we must perform the same operation on the other side to maintain the equality. It's like a balanced scale – if you add weight to one side, you must add the same weight to the other to keep it level. We'll be using addition, subtraction, multiplication, and division to move terms and simplify the equation until 'x' stands alone, revealing its value.
Step 1: Combine 'x' Terms
The first strategic move in solving 1.8 - 3.7x = -4.2x + 0.3 is to gather all the terms containing 'x' onto one side of the equation. This simplifies the problem by reducing the number of places 'x' appears. We have '-3.7x' on the left side and '-4.2x' on the right side. To bring them together, we can choose to move either term. A common preference is to move the term with the smaller coefficient to avoid negative numbers initially, but either way works. Let's decide to move the '-4.2x' from the right side to the left side. To do this, we perform the opposite operation: adding '4.2x' to both sides of the equation.
So, we have:
1.8 - 3.7x + 4.2x = -4.2x + 0.3 + 4.2x
On the right side, '-4.2x' and '+4.2x' cancel each other out, leaving us with just '0.3'. On the left side, we combine the 'x' terms: '-3.7x + 4.2x'. This calculation is 4.2 - 3.7, which equals 0.5. So, the left side becomes '1.8 + 0.5x'.
Our equation now looks like this:
1.8 + 0.5x = 0.3
See? We've successfully reduced the complexity by consolidating our 'x' terms. This is a crucial step that brings us closer to isolating 'x'. It's all about strategic simplification, making the equation more manageable with each algebraic maneuver.
Step 2: Isolate the 'x' Term
Now that our 'x' terms are combined on one side, the next logical step in solving 1.8 + 0.5x = 0.3 is to get the term containing 'x' completely by itself on one side. Currently, we have '1.8' added to '0.5x' on the left side. To isolate '0.5x', we need to eliminate the '+1.8'. We do this by performing the opposite operation: subtracting '1.8' from both sides of the equation. This maintains the balance of the equation.
Applying this, we get:
1.8 + 0.5x - 1.8 = 0.3 - 1.8
On the left side, '+1.8' and '-1.8' cancel each other out, leaving us with just '0.5x'. On the right side, we perform the subtraction: '0.3 - 1.8'. This results in '-1.5'.
So, our equation is now simplified to:
0.5x = -1.5
We're in the home stretch! The 'x' term is now isolated. The next step will be to find the value of a single 'x'. This process of isolating the variable term is fundamental in algebra, and it's exactly what we need to do to prepare for the final step of finding 'x'.
Step 3: Solve for 'x'
We've reached the final stage in solving 0.5x = -1.5. Our goal is to find the value of a single 'x', not '0.5x'. Currently, 'x' is being multiplied by 0.5. To isolate 'x', we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by the coefficient of 'x', which is 0.5.
Let's perform the division:
(0.5x) / 0.5 = (-1.5) / 0.5
On the left side, '0.5' divided by '0.5' equals 1, leaving us with '1x', or simply 'x'. On the right side, we need to calculate '-1.5 divided by 0.5'. Dividing a negative number by a positive number results in a negative number. So, 1.5 divided by 0.5 is 3. Therefore, '-1.5 divided by 0.5' is '-3'.
This gives us our final answer:
x = -3
Congratulations! You've successfully navigated the steps to solve for 'x'. You've combined like terms, isolated the variable term, and finally solved for 'x'. Remember, the key is to apply inverse operations consistently to both sides of the equation to maintain equality. This systematic approach works for a wide range of algebraic problems.
Verification
To ensure our solution is correct, let's plug 'x = -3' back into the original equation: 1.8 - 3.7x = -4.2x + 0.3.
Left Side: 1.8 - 3.7*(-3) 1.8 - (-11.1) 1.8 + 11.1 = 12.9
Right Side: -4.2*(-3) + 0.3 12.6 + 0.3 = 12.9
Since the left side (12.9) equals the right side (12.9), our solution x = -3 is correct!
Conclusion
We've successfully unraveled the mystery of the equation 1.8 - 3.7x = -4.2x + 0.3, finding that the value of 'x' is -3. This journey through solving linear equations highlights the power of consistent application of algebraic principles. By combining like terms, isolating the variable, and performing inverse operations, we can tackle complex-looking problems with confidence. Remember, practice makes perfect! The more equations you solve, the more intuitive these steps will become. Keep exploring the fascinating world of mathematics, and don't be afraid to challenge yourself with new problems.
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