Solving For X: Easy Math Problems

Hey math whizzes and curious minds! Today, we're diving into the exciting world of algebra to tackle a couple of problems where we need to solve for X. Don't worry, guys, these aren't super complicated, and we'll break them down step-by-step so everyone can follow along. Algebra can seem intimidating, but it's really just about figuring out an unknown value, and that's kind of like solving a puzzle! We'll be looking at two specific equations: $3-.05 X =.2 X -1.25$ and rac{x-8}{7}+x=4. These types of problems are fundamental in mathematics and pop up in all sorts of real-world scenarios, from calculating discounts to understanding scientific formulas. So, grab your pencils, get comfortable, and let's get ready to unravel these algebraic mysteries together!

Problem 19: Decoding $3-.05 X =.2 X -1.25$

Alright team, let's get down to business with our first problem: $3-.05 X =.2 X -1.25$. Our main goal here is to isolate the variable X on one side of the equation. To do this, we need to get all the terms with X on one side and all the constant terms (the numbers without X) on the other. Think of the equals sign as a balancing scale; whatever we do to one side, we must do to the other to keep it balanced.

First off, let's gather our X terms. I see $-.05X$ on the left and $.2X$ on the right. To move the $-.05X$ to the right, we can add $.05X$ to both sides. This cancels it out on the left and gives us $.2X + .05X$ on the right. So, our equation now looks like this: $3 = .25X - 1.25$. See? We're already making progress! Now, we need to get those constant terms together. We have a '3' on the left and a '-1.25' on the right. To move the '-1.25' to the left, we'll do the opposite and add $1.25$ to both sides. This gives us $3 + 1.25$ on the left, which equals $4.25$. Our equation transforms into $4.25 = .25X$.

We're almost there! The last step to solve for X is to get X all by itself. Right now, it's being multiplied by $.25$. To undo multiplication, we use division. So, we'll divide both sides of the equation by $.25$. On the left, $4.25 / .25$ equals $17$. On the right, $.25X / .25$ just leaves us with X. Therefore, our solution is $X = 17$.

To double-check our work, we can plug $X=17$ back into the original equation: $3 - .05(17) = .2(17) - 1.25$. Let's calculate: $3 - .85 = 3.4 - 1.25$. Simplifying both sides gives us $2.15 = 2.15$. It matches! High five, we got it right!

Problem 20: Tackling rac{x-8}{7}+x=4

Now, let's move on to our second challenge: rac{x-8}{7}+x=4. This one looks a little different because it has a fraction. The key here is to get rid of that fraction to make the equation easier to work with. The denominator of our fraction is 7. To eliminate it, we're going to multiply every single term in the equation by 7. Remember, whatever we do to one part, we do to all parts to keep things fair and balanced.

So, let's multiply each term by 7: 7 imes rac{x-8}{7} + 7 imes x = 7 imes 4.

When we multiply the first term, the 7 in the numerator and the 7 in the denominator cancel each other out, leaving us with just $x-8$. The second term becomes $7x$. And the right side of the equation, $7 imes 4$, is $28$. Our equation now simplifies to: $x - 8 + 7x = 28$.

See how much cleaner that looks? Now we can combine the like terms. We have an 'x' and a '7x' on the left side. Adding them together gives us $8x$. So, the equation becomes $8x - 8 = 28$.

Our next move is to get the $8x$ term by itself. We have a '-8' on the left. To move it, we add 8 to both sides of the equation. This gives us $8x = 28 + 8$, which simplifies to $8x = 36$.

Finally, to solve for X, we need to undo the multiplication. X is currently being multiplied by 8. So, we divide both sides by 8. This gives us X = rac{36}{8}.

Now, we can simplify this fraction. Both 36 and 8 are divisible by 4. $36 ext{ divided by } 4 ext{ is } 9$, and $8 ext{ divided by } 4 ext{ is } 2$. So, our simplified answer is X = rac{9}{2}, or as a decimal, $X = 4.5$.

Let's check our answer by plugging $X = 4.5$ back into the original equation: rac{4.5 - 8}{7} + 4.5 = 4. Calculating the numerator: $4.5 - 8 = -3.5$. So we have rac{-3.5}{7} + 4.5 = 4. Now, $-3.5$ divided by $7$ is $-0.5$. So, $-0.5 + 4.5 = 4$. And indeed, $-0.5 + 4.5$ equals $4$. Wow, it checks out! Another puzzle solved!

Why Solving for X Matters

So, why do we bother with these algebraic equations and learning to solve for X? Well, guys, it's more than just homework! Understanding how to manipulate equations and find unknown values is a crucial skill that builds a foundation for more advanced math and science. Think about it: whenever engineers design a bridge, scientists conduct experiments, or even when you're trying to figure out how much paint you need for a room, you're essentially using algebraic thinking. You have knowns and unknowns, and you need to find the missing piece.

These problems, like the ones we just solved, teach us systematic thinking and problem-solving strategies. We learned the importance of performing operations on both sides of an equation to maintain balance, how to combine like terms, and how to use inverse operations (like addition and subtraction, or multiplication and division) to isolate variables. These aren't just math rules; they are logical steps that can be applied to solve problems in any field. The ability to approach a complex situation, break it down into manageable parts, and systematically work towards a solution is invaluable. Whether you're debugging code, planning a budget, or figuring out the best route to take, the underlying principles of algebraic problem-solving are at play. Keep practicing, and you'll find that solving for X, and many other things, becomes second nature!