Complete Table & Graph Y = 5 - 2x: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a classic algebra problem: completing a table of values for a linear equation and then graphing it. We'll be working with the equation y = 5 - 2x, and trust me, it's easier than it sounds. So, grab your pencils, and let's get started!

Understanding Linear Equations

Before we jump into the problem, let's quickly recap what a linear equation is. A linear equation is an equation that, when graphed, forms a straight line. The general form of a linear equation is y = mx + c, where 'm' represents the slope of the line and 'c' represents the y-intercept (the point where the line crosses the y-axis). In our case, the equation y = 5 - 2x is indeed a linear equation, just written in a slightly different order. We can rewrite it as y = -2x + 5 to clearly see that the slope is -2 and the y-intercept is 5.

Linear equations are fundamental in mathematics and have numerous applications in real-world scenarios, from calculating distances and speeds to modeling financial trends. Understanding how to work with them is a crucial skill in algebra and beyond. The beauty of linear equations lies in their simplicity and predictability. Because they form straight lines, we can easily visualize and analyze their behavior. By finding just a few points that satisfy the equation, we can accurately draw the entire line. This is precisely what we'll be doing in this tutorial.

When approaching a linear equation, remember that each point on the line represents a solution to the equation. This means that if we plug the x and y coordinates of a point on the line into the equation, it will hold true. This is the key idea behind creating a table of values – we're simply finding several points that satisfy the equation. Mastering linear equations opens doors to more advanced mathematical concepts. They form the building blocks for systems of equations, inequalities, and even calculus. So, take your time, practice, and you'll find that they become second nature. This step-by-step guide will give you a solid foundation for working with these important equations. Let’s move on to how to complete the table for the linear equation in this example.

(a) Completing the Table for y = 5 - 2x

Our first task is to complete the table of values for the relation y = 5 - 2x within the range of -3 ≤ x ≤ 4. This means we need to find the corresponding 'y' values for each 'x' value given in the table. Let's break it down step by step:

Here's the table we need to complete:

We already have some values filled in, which is a great starting point. We'll use the equation y = 5 - 2x to calculate the missing 'y' values for each 'x' value.

1. For x = -2:

   • Substitute x = -2 into the equation: y = 5 - 2(-2)
   • Simplify: y = 5 + 4 = 9

2. For x = -1:

   • Substitute x = -1 into the equation: y = 5 - 2(-1)
   • Simplify: y = 5 + 2 = 7

3. For x = 1:

   • Substitute x = 1 into the equation: y = 5 - 2(1)
   • Simplify: y = 5 - 2 = 3

4. For x = 3:

   • Substitute x = 3 into the equation: y = 5 - 2(3)
   • Simplify: y = 5 - 6 = -1

Now, let's fill in the completed table:

And there you have it! We've successfully completed the table of values for the relation y = 5 - 2x. This table gives us a set of coordinate pairs (x, y) that we can use to plot the graph of the equation. Each pair represents a point on the line. Completing the table accurately is crucial for graphing the line correctly. A single mistake in the calculation can lead to an incorrect graph. Double-checking your work is always a good practice. Now that we have our table filled in, we are well-prepared to move on to the next stage, which is graphing the equation. The table acts as a roadmap for drawing the line, guiding us to plot the points accurately on the coordinate plane.

(b) Graphing the Equation

Now that we have our completed table of values, we can move on to graphing the equation y = 5 - 2x. We'll be using a scale of 2 cm to represent 1 unit on both axes. This means that each 2 cm increment on the graph paper will correspond to a value of 1 in either the x or y direction. This scale is important because it helps us create a clear and accurate representation of the line.

Setting up the Axes

First, we need to draw our x and y axes. Since our x-values range from -3 to 4 and our y-values range from -3 to 11, we need to make sure our graph is large enough to accommodate these values. Draw a horizontal line for the x-axis and a vertical line for the y-axis, making sure they intersect at the origin (0, 0). Remember to label the axes as 'x' and 'y'.

Next, we need to mark the scale on each axis. Using our 2 cm per unit scale, mark increments of 1 on both axes. For example, 2 cm to the right of the origin will be '1' on the x-axis, 4 cm to the right will be '2', and so on. Similarly, 2 cm above the origin will be '1' on the y-axis, 4 cm above will be '2', and so on. Remember to mark negative values as well, extending the scale to the left and below the origin.

Plotting the Points

Now comes the exciting part – plotting the points from our table! Each (x, y) pair in the table represents a point on the graph. To plot a point, find the corresponding x-value on the x-axis and the corresponding y-value on the y-axis. Then, mark the point where these two values intersect. Let's plot a few points as examples:

• Point (-3, 11): Find -3 on the x-axis and 11 on the y-axis. Mark the point where these two lines meet.
• Point (-2, 9): Find -2 on the x-axis and 9 on the y-axis. Mark the point where they meet.
• Point (0, 5): Find 0 on the x-axis (the origin) and 5 on the y-axis. Mark the point.

Continue plotting all the points from your completed table. You should have eight points in total.

Plotting points accurately is the cornerstone of graphing linear equations. Each point must be precisely located on the coordinate plane, reflecting its corresponding x and y values. A slight deviation can lead to an inaccurate representation of the line. Use a ruler to ensure your axes are straight and your scale is consistent. Taking the time to plot each point carefully will pay off when you connect them to form the line. As you plot each point, mentally check if it aligns with the overall trend suggested by the previous points. This helps you catch any potential errors in your calculations or plotting.

Drawing the Line

Once you've plotted all the points, you should notice that they form a straight line. This is because we're graphing a linear equation! Now, take a ruler and carefully draw a straight line that passes through all the plotted points. Make sure the line extends beyond the points on both ends, indicating that the line continues infinitely in both directions.

And that's it! You've successfully graphed the equation y = 5 - 2x. The line you've drawn represents all the possible solutions to the equation. Any point on the line will satisfy the equation when its x and y coordinates are plugged in. Drawing a precise line requires a steady hand and a sharp pencil. A well-drawn line not only represents the equation accurately but also makes the graph easier to interpret. If you notice that your points don't quite form a straight line, it might indicate an error in your calculations or plotting. Go back and double-check your work to ensure accuracy.

Checking Your Graph

A good practice is to check your graph by picking a point on the line (that wasn't in your original table) and plugging its coordinates into the equation. If the equation holds true, your graph is likely correct. For example, you might pick the point (-4, 13), which should lie on the line if you've drawn it correctly. Substituting these values into the equation gives us: 13 = 5 - 2(-4), which simplifies to 13 = 5 + 8, and finally, 13 = 13. Since the equation holds true, we can be confident in our graph.

Checking your work is a crucial step in any mathematical problem. It helps you catch errors and reinforces your understanding of the concepts. By verifying your graph, you're ensuring that your visual representation accurately reflects the algebraic equation. This process of connecting the algebraic and graphical representations is a fundamental aspect of mathematics.

Key Takeaways

Let's recap the key steps we've covered in this tutorial:

1. Completing the Table: We substituted x-values into the equation y = 5 - 2x to find the corresponding y-values and fill in the table.
2. Setting up the Axes: We drew and labeled the x and y axes, using a scale of 2 cm to represent 1 unit.
3. Plotting the Points: We plotted the (x, y) pairs from the table onto the graph.
4. Drawing the Line: We used a ruler to draw a straight line through the plotted points.
5. Checking Your Graph: We learned the importance of checking your graph by picking a point on the line and plugging its coordinates into the equation.

By following these steps, you can confidently complete tables and graph linear equations. Remember, practice makes perfect! The more you work with linear equations, the more comfortable and proficient you'll become.

Practice Makes Perfect

To solidify your understanding, try graphing other linear equations. You can create your own equations or find examples in your textbook. Experiment with different scales on the axes to see how they affect the appearance of the graph. The ability to confidently graph linear equations is a valuable skill that will serve you well in your mathematical journey. Continue to explore different types of equations and their graphs to broaden your understanding of mathematical concepts.

Conclusion

Congratulations! You've successfully learned how to complete a table of values and graph the linear equation y = 5 - 2x. By understanding these fundamental concepts, you're well-equipped to tackle more complex algebraic problems. Keep practicing, and you'll be a graphing pro in no time!

For more information on linear equations and graphing, you can visit Khan Academy's Linear Equations section. Happy graphing!