Equation Of A Line: Slope 5, Point (5, 19) - Solved!

Hey everyone! Let's dive into a classic algebra problem: finding the equation of a line. Specifically, we're going to figure out the equation of a line that has a slope of 5 and passes through the point (5, 19). Don't worry, it's not as scary as it sounds! We'll break it down step-by-step so you can totally nail it. We will be using the slope-intercept form, which is super useful and you'll find it popping up all over the place in math. The key here is understanding a few core concepts and then just plugging in the numbers. This is a fundamental concept in mathematics and is a building block for more complex topics like calculus and linear algebra. Grasping this now will give you a significant advantage in future math endeavors. Let's get started, shall we?

Understanding the Slope-Intercept Form

Okay, before we jump into the problem, let's make sure we're all on the same page about the slope-intercept form. This is the standard way to write the equation of a line, and it looks like this: y = mx + b. Now, what do all those letters mean? Well:

• y and x are the variables. They represent the coordinates of any point on the line. Think of them as the general 'x' and 'y' values.
• m is the slope of the line. This tells us how steep the line is. A slope of 5, like in our problem, means the line goes up 5 units for every 1 unit it moves to the right.
• b is the y-intercept. This is the point where the line crosses the y-axis (the vertical axis). It's the 'y' value when 'x' is zero.

So, basically, the slope-intercept form gives us a clear picture of a line: its steepness (m) and where it crosses the y-axis (b). Got it? Great! Now, we're ready to tackle our problem. This form is incredibly useful because it allows us to easily visualize the line and understand its properties. Once you understand this form, working with linear equations becomes much more manageable. Remember, the key is understanding what each part of the equation represents. The slope-intercept form is used extensively in various fields such as physics, engineering, and computer science, to model linear relationships.

Finding the Equation: Step-by-Step

Alright, let's get down to business. We know the slope (m) is 5, and we have a point on the line, (5, 19). Our goal is to find the equation in the form y = mx + b. We already know m, so we just need to figure out b (the y-intercept).

Here’s how we can do it:

1. Start with the Slope-Intercept Form: Write down y = mx + b. This is our base equation.
2. Plug in the Slope: We know m = 5, so substitute that into the equation: y = 5x + b.
3. Use the Point (5, 19): This point tells us that when x = 5, y = 19. Substitute these values into the equation: 19 = 5(5) + b.
4. Solve for b: Simplify the equation: 19 = 25 + b. Then, subtract 25 from both sides: 19 - 25 = b. This gives us b = -6.
5. Write the Final Equation: Now that we know m = 5 and b = -6, we can write the final equation: y = 5x - 6. Boom! We did it! This step-by-step approach ensures that you understand the process and can apply it to any similar problem. Remember, the point (5, 19) lies on the line, and its coordinates satisfy the equation we found. We used the given point to find the y-intercept, which is a crucial step in this process. By substituting the values of x and y from the point, we could isolate b and find its value. The y-intercept helps define the position of the line on the y-axis, and in this case, the line intersects the y-axis at -6.

Verifying the Solution

It's always a good idea to double-check our work. We can do this by plugging the point (5, 19) back into our equation, y = 5x - 6. Let's see if it works:

• Substitute x = 5 and y = 19 into the equation: 19 = 5(5) - 6.
• Simplify: 19 = 25 - 6.
• Further simplification gives us 19 = 19.

The equation holds true! This means that the point (5, 19) does lie on the line defined by our equation, and we've successfully solved the problem. Seeing that both sides of the equation are equal confirms that our calculations were accurate. This step is essential to ensure that there were no errors in the process and provides a sense of confidence in the solution. This is an important step to make sure the solution is correct. By verifying the solution, you not only confirm your answer but also reinforce your understanding of the concepts involved. It's a great habit to get into. You'll avoid making silly mistakes and boost your confidence in your math skills. Moreover, understanding how to verify your answers is crucial in real-world scenarios, where accuracy is paramount. This verification step also allows you to catch any potential errors and refine your problem-solving abilities.

Conclusion: You Got This!

So there you have it! We've successfully found the equation of a line with a slope of 5 that passes through the point (5, 19). The equation in slope-intercept form is y = 5x - 6. Remember, the key is to understand the slope-intercept form, plug in the known values, and solve for the unknown. Practice makes perfect, so try some similar problems on your own. You'll be a pro in no time! Keep practicing, and you'll find that these types of problems become easier with each attempt. Mastering this concept will serve as a solid foundation for more complex mathematical concepts. Don’t be afraid to ask for help or check your work. Good luck, and happy math-ing!