Solve For X: Find The Outlier Equation

Hey math whizzes! Let's dive into a fun little algebra puzzle. We've got four equations, each featuring the variable x, and our mission is to pinpoint the one that gives us a different solution for x compared to the others. Sounds like a blast, right? We'll break down each equation step by step, solving for x and then compare our answers. By doing this, we'll not only solve the problem, but also sharpen our algebra skills. Are you ready to crack some equations?

Equation Breakdown and Solutions

Let's tackle each equation individually, shall we? We'll go through them one by one, solving for x with meticulous care. This approach will allow us to compare the answers and identify the outlier. Remember, the goal is not just to find the answer, but to understand the process. Grasping the how and why behind each step is what will cement our understanding of algebra. So, let's roll up our sleeves and get started!

Equation A: $-\frac{7}{8} x - \frac{3}{4} = 20$

First up, we have the equation $-\frac{7}{8} x - \frac{3}{4} = 20$. Our aim is to isolate x on one side of the equation. To begin, we'll add $\frac{3}{4}$ to both sides of the equation. This gets rid of the $-\frac{3}{4}$ on the left side, leaving us with: $-\frac{7}{8} x = 20 + \frac{3}{4}$. Now, we need to add 20 and $\frac{3}{4}$. To do this, we can write 20 as $\frac{80}{4}$ so we can have common denominators. Thus, $-\frac{7}{8} x = \frac{80}{4} + \frac{3}{4}$. Combining the fractions gives us $-\frac{7}{8} x = \frac{83}{4}$.

Next, to isolate x, we'll multiply both sides of the equation by the reciprocal of $-\frac{7}{8}$, which is $-\frac{8}{7}$. This gives us: $x = \frac{83}{4} \cdot -\frac{8}{7}$. Simplifying, we get $x = -\frac{83 \cdot 8}{4 \cdot 7}$. Further simplifying, we have $x = -\frac{83 \cdot 2}{7}$. Therefore, $x = -\frac{166}{7}$. So, the solution for Equation A is x = -166/7. This result is crucial, so let's mark it and proceed. Now that we have calculated this, let's move forward.

Equation B: $\frac{3}{4} + \frac{7}{8} x = -20$

Let's turn our attention to Equation B: $\frac{3}{4} + \frac{7}{8} x = -20$. Similar to before, our initial step is to isolate the term containing x. We start by subtracting $\frac{3}{4}$ from both sides. This leads us to: $\frac{7}{8} x = -20 - \frac{3}{4}$. We can rewrite -20 as -80/4 so we can combine like terms. Hence, $\frac{7}{8} x = -\frac{80}{4} - \frac{3}{4}$. Combining these, we obtain $\frac{7}{8} x = -\frac{83}{4}$.

To isolate x, we multiply both sides of the equation by the reciprocal of $\frac{7}{8}$, which is $\frac{8}{7}$. Doing so, we get $x = -\frac{83}{4} \cdot \frac{8}{7}$. Then, we simplify by multiplying. Hence, $x = -\frac{83 \cdot 8}{4 \cdot 7}$. Further simplifying this, we end up with $x = -\frac{83 \cdot 2}{7}$. Thus, $x = -\frac{166}{7}$. The solution for Equation B is also x = -166/7. It seems that we have found a match. As we progress, we will see if we can locate the equation that is different from the others. Remember, every step is a clue in this mathematical investigation, and we must pay attention to every detail.

Equation C: $-7(\frac{1}{8}) x - \frac{3}{4} = 20$

Alright, let's focus on Equation C: $-7(\frac{1}{8}) x - \frac{3}{4} = 20$. We can rewrite $-7(\frac{1}{8})$ as $-\frac{7}{8}$. So now our equation is $-\frac{7}{8} x - \frac{3}{4} = 20$. This equation is the same as Equation A! Since this is the same as equation A, we know the answer is still -166/7. We will proceed to the next equation to determine if it is the outlier.

Equation D: $-\frac{7}{8}(-\frac{8}{7})x$

Finally, let's solve Equation D: $-\frac{7}{8} (-\frac{8}{7}) x$. First, we have to deal with the multiplication. $-\frac{7}{8} \cdot -\frac{8}{7} = 1$. This means the equation is: $1x = 20$, or more simply, $x=20$. We have found our outlier! After solving each equation, we found that equations A, B, and C all have a solution for x of -166/7, and equation D has a solution of x=20. This indicates that Equation D is the one that results in a different value of x. The math game is won!

Conclusion: The Outlier Revealed!

Well, guys, we've successfully navigated through the maze of equations. We've solved for x in each one and compared our results. The equations A, B, and C all gave us the same solution, while Equation D stood out with a different answer. Therefore, we can definitively say that Equation D is the outlier. It's awesome to see how slight variations in the structure of an equation can lead to drastically different results. This exercise not only strengthens our algebra skills but also highlights the importance of precision in mathematical problem-solving. Keep up the awesome work!

Tips for Solving Equations

• Always Isolate the Variable: The fundamental rule is to get the variable by itself on one side of the equation. This involves performing inverse operations (addition/subtraction, multiplication/division) to both sides.
• Simplify Step by Step: Break down complex equations into smaller, manageable steps. This reduces the chance of errors and makes the process more clear.
• Double-Check Your Work: After finding a solution, substitute it back into the original equation to verify that it holds true. This is an excellent way to catch any mistakes.

We did it! We have solved the equation and found the outlier equation. Keep practicing, and you'll become a pro in no time! Keep exploring, keep questioning, and above all, keep having fun with math! See you next time, math enthusiasts!