Solving Equations And Proportions A Step-by-Step Guide

The letters $a$ and $b$ represent nonzero constants. Solve $ax + b = 27$ for $x$.

Hey guys! Let's dive into solving this equation. The goal here is to isolate $x$ on one side of the equation. We're given the equation $ax + b = 27$, where $a$ and $b$ are nonzero constants. Don't let the constants freak you out; we'll treat them just like numbers.

To get $x$ by itself, we first need to get rid of that $+ b$ term. How do we do that? Simple! We subtract $b$ from both sides of the equation. This keeps the equation balanced, which is super important. So, we have:

$ax + b - b = 27 - b$

This simplifies to:

$ax = 27 - b$

Now, we need to get rid of the $a$ that's multiplying $x$. To do this, we divide both sides by $a$. Remember, $a$ is a nonzero constant, so we're not dividing by zero, which is a big no-no in math. We get:

$\frac{ax}{a} = \frac{27 - b}{a}$

This simplifies to:

$x = \frac{27 - b}{a}$

And there you have it! We've solved for $x$. The solution is $x = \frac{27 - b}{a}$. This is the key step, guys, in isolating the variable. Understanding this process will help you tackle all sorts of linear equations. Remember, it’s all about performing the same operations on both sides to maintain balance.

The solution is $x=\frac{27-b}{a}$. This solution represents the value of $x$ that satisfies the original equation $ax + b = 27$. By isolating $x$, we've essentially unwrapped the equation to reveal its solution. This method is crucial not just for this specific problem but for solving any linear equation. The core concept is to reverse the operations applied to $x$, ensuring each step maintains the equation's balance. This foundation is incredibly valuable as you progress in mathematics and encounter more complex equations and problems. Keep practicing these steps, and you'll find solving for variables becomes second nature. The beauty of algebra lies in its systematic approach, and mastering these basics opens the door to more advanced topics. So, keep up the great work, and remember, every equation you solve is a step forward in your mathematical journey!

You are 5 feet 6 inches tall and cast a shadow 4 feet long. At the same time, a nearby flagpole casts a shadow 22 feet long. How tall is the flagpole?

Okay, let's tackle this flagpole problem! This is a classic example of using proportions to solve real-world problems. The key here is that we can set up a proportion based on similar triangles. When the sun shines, you, the flagpole, and your respective shadows form two similar right triangles.

First, we need to make sure our units are consistent. You're given as 5 feet 6 inches tall. Let's convert that entirely to inches. There are 12 inches in a foot, so 5 feet is $5 * 12 = 60$ inches. Add the extra 6 inches, and you're $60 + 6 = 66$ inches tall. Your shadow is 4 feet long, which is $4 * 12 = 48$ inches.

Now, the flagpole's shadow is 22 feet long, which is $22 * 12 = 264$ inches. Let's call the height of the flagpole $h$ (in inches). We can set up a proportion like this:

$\frac{\text{Your Height}}{\text{Your Shadow Length}} = \frac{\text{Flagpole Height}}{\text{Flagpole Shadow Length}}$

Plugging in the values, we get:

$\frac{66}{48} = \frac{h}{264}$

To solve for $h$, we can cross-multiply. Cross-multiplication is a handy trick for solving proportions. We multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa:

$66 * 264 = 48 * h$

This gives us:

$17424 = 48h$

Now, to isolate $h$, we divide both sides by 48:

$\frac{17424}{48} = h$

$h = 363$

So, the flagpole is 363 inches tall. But let's convert that back to feet and inches to make it more relatable. We divide 363 by 12:

$363 \div 12 = 30$ with a remainder of 3

This means the flagpole is 30 feet and 3 inches tall. Isn't that cool? We used proportions to find the height of the flagpole without actually climbing it!

The flagpole problem beautifully illustrates the power of proportions and similar triangles in solving practical problems. The core of the solution lies in recognizing the geometric similarity between your shadow and the flagpole's shadow, allowing us to set up a proportional relationship between the heights and shadow lengths. By meticulously converting units to ensure consistency and applying the cross-multiplication technique, we were able to determine the flagpole's height. This problem emphasizes the importance of proportional reasoning in mathematics and its applications in everyday scenarios. Understanding similar triangles and proportions not only enhances problem-solving skills but also provides a framework for tackling a variety of real-world challenges, from architecture and engineering to navigation and cartography. Keep practicing these concepts, and you'll find they become increasingly intuitive and powerful tools in your mathematical arsenal. Remember, the key is to break down complex problems into simpler, proportional relationships, making them much easier to solve.

So, guys, we've walked through solving a linear equation for $x$ and tackled a proportion problem using similar triangles. These are fundamental skills in math, and mastering them will set you up for success in more advanced topics. Remember, the key is to break down problems into smaller, manageable steps and to always keep the equation balanced. Keep practicing, and you'll become math whizzes in no time! Keep up the great work, and never stop exploring the fascinating world of mathematics. Each problem you solve is a step forward in your journey, and the more you practice, the more confident and skilled you'll become. Remember, math isn't just about numbers and equations; it's about developing critical thinking and problem-solving skills that will serve you well in all aspects of life. So, embrace the challenges, celebrate your successes, and keep pushing your boundaries. The world of mathematics is vast and exciting, and there's always something new to discover.