Finding The Linear Equation From A Table A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of linear equations. We've got a table of values, and our mission, should we choose to accept it, is to crack the code and figure out the linear equation that governs this relationship. So, buckle up, math enthusiasts, let's get started!

Unveiling the Linear Equation from a Table

Our primary goal here is to find the linear equation that perfectly describes the relationship between x and y in the given table. We're looking for an equation in the form of y = mx + b, where m represents the slope (the rate of change) and b represents the y-intercept (where the line crosses the y-axis). To decipher this linear code, we'll embark on a step-by-step journey, carefully analyzing the data points provided in the table and skillfully piecing together the equation that connects them all. This process involves a keen eye for patterns, a dash of algebraic prowess, and a whole lot of logical deduction. We'll start by calculating the slope, which essentially tells us how much y changes for every unit change in x. Then, we'll use this slope, along with one of the points from the table, to pinpoint the y-intercept. Once we have both the slope and the y-intercept, we can confidently construct the linear equation that rules this numerical landscape. So, let's roll up our sleeves and dive into the nitty-gritty of slope calculation and y-intercept determination. Are you ready to transform these table entries into a sleek, elegant linear equation? Let's do this!

Step 1 Spotting the Pattern and Calculating the Slope

First things first, let's take a close look at the table. We've got our x values and their corresponding y values. To calculate the slope, we need to see how y changes as x changes. Remember, the slope (m) is the change in y divided by the change in x. Think of it like this: if x increases by a certain amount, how much does y increase or decrease? This change is the key to unlocking the slope. We can pick any two points from the table to do this calculation. For example, let's take the first two points: (-18, 1) and (-14, 9). Now, let's calculate the change in y which is 9 - 1 = 8, and the change in x which is -14 - (-18) = 4. So, the slope (m) is 8 / 4 = 2. What this means is that for every 1 unit increase in x, y increases by 2 units. But, don't just take our word for it! Let's verify this slope with another pair of points, just to be absolutely sure. Let's choose (-10, 17) and (-6, 25). The change in y is 25 - 17 = 8, and the change in x is -6 - (-10) = 4. Guess what? The slope is still 8 / 4 = 2! This consistency is a strong indicator that we're on the right track to unveiling the linear equation. The consistent slope confirms that the relationship between x and y is indeed linear. Now that we have confidently determined the slope, we're one step closer to our ultimate goal. With the slope firmly in our grasp, we can proceed to the next crucial step: finding the y-intercept. This will be the final piece of the puzzle, allowing us to construct the complete linear equation. So, let's forge ahead, armed with our newfound slope, and conquer the y-intercept!

Step 2 Finding the Y-Intercept

Okay, guys, we've nailed down the slope (m), which is 2. Now, it's time to hunt for the y-intercept (b). Remember, the y-intercept is the point where the line crosses the y-axis, and it's the y value when x is 0. To find b, we can use the slope-intercept form of a linear equation, which is y = mx + b, and plug in the slope we just found, along with the coordinates of any point from our table. Let's use the point (-18, 1). Substituting these values into the equation, we get 1 = 2 * (-18) + b. Now, it's just a matter of solving for b. First, we multiply 2 by -18, which gives us -36. So, our equation becomes 1 = -36 + b. To isolate b, we add 36 to both sides of the equation: 1 + 36 = b. This simplifies to 37 = b. So, we've found our y-intercept! b is 37. But, just to be extra sure (because in math, it's always good to double-check), let's use another point from the table and see if we get the same y-intercept. Let's use the point (-14, 9). Plugging this into the equation y = mx + b, we get 9 = 2 * (-14) + b. Multiplying 2 by -14 gives us -28, so the equation becomes 9 = -28 + b. Adding 28 to both sides, we get 9 + 28 = b, which simplifies to 37 = b. Hooray! We got the same y-intercept again! This gives us even more confidence that we're on the right track. Now that we have both the slope (m = 2) and the y-intercept (b = 37), we have all the pieces we need to construct the linear equation. We're almost there, guys! The final step is just putting it all together in the correct format. Let's do it!

Step 3 Writing the Linear Equation

Alright, mathletes, we've reached the final stretch! We've successfully calculated the slope (m = 2) and the y-intercept (b = 37). Now, it's time to write the linear equation in its full glory. Remember, the slope-intercept form of a linear equation is y = mx + b. We simply need to substitute our values for m and b into this equation. So, replacing m with 2 and b with 37, we get y = 2x + 37. And there you have it! This is the linear equation that represents the relationship between x and y in the table. But before we pop the champagne and celebrate our victory, let's do one last quick check to make sure our equation is spot-on. We can do this by plugging in a few x values from the table into our equation and seeing if we get the corresponding y values. For example, let's use x = -18. Plugging this into our equation, we get y = 2 * (-18) + 37, which simplifies to y = -36 + 37, and that equals 1. This matches the y value in our table for x = -18! Let's try another one. Let's use x = -10. Plugging this in, we get y = 2 * (-10) + 37, which simplifies to y = -20 + 37, and that equals 17. Again, this matches the table! These checks further solidify our confidence that y = 2x + 37 is indeed the correct linear equation. Woohoo! We've successfully decoded the linear relationship hidden within the table. Give yourselves a pat on the back, guys! You've demonstrated your mathematical prowess and your ability to crack the code of linear equations. Now, go forth and conquer more mathematical challenges!

Final Answer

The linear equation that gives the rule for the table is:

y = 2x + 37

Great job, everyone! You've mastered the art of finding linear equations from tables. Keep up the awesome work!