Completing The Table For Y = 2x - 5 A Step-by-Step Guide
Hey guys! Today, we're diving into a super important concept in algebra: completing tables for linear equations. Specifically, we're going to tackle the equation y = 2x - 5. This is a classic example that will help you understand how to find ordered pairs and visualize linear relationships. So, grab your pencils, and let's get started!
Understanding the Basics of Linear Equations
Before we jump into the table, let's quickly review what a linear equation is. In simple terms, a linear equation is an equation that, when graphed, forms a straight line. The equation y = 2x - 5 is in slope-intercept form (y = mx + b), where:
- m represents the slope of the line.
- b represents the y-intercept (the point where the line crosses the y-axis).
In our equation, y = 2x - 5, the slope (m) is 2, and the y-intercept (b) is -5. This means that for every 1 unit we move to the right on the graph, the line goes up 2 units. Also, the line crosses the y-axis at the point (0, -5).
Understanding these basics is crucial because it helps us predict the behavior of the line and accurately fill in the table. We'll be using the equation to find the corresponding y-values for given x-values, creating ordered pairs that represent points on this line.
Setting Up the Table: X and Y Values
Okay, now let’s talk about the table we’re going to complete. Tables are super helpful for organizing information and seeing patterns. In this case, our table has three columns:
- x: This column lists the x-values we'll be using. These are the inputs for our equation.
- y: This column is where we'll calculate the corresponding y-values using the equation y = 2x - 5. These are the outputs.
- Ordered Pair: This column will list the ordered pairs (x, y) that we find. Remember, an ordered pair represents a specific point on the graph of the line.
The x-values provided are: -2, -1, 0, 1, and 2. Our mission is to plug each of these x-values into the equation y = 2x - 5 and solve for y. Once we have both the x and y values, we can write them as an ordered pair. This process might seem a bit tedious, but trust me, it becomes second nature with practice!
Step-by-Step Guide to Completing the Table
Alright, let's get down to business and fill in this table! We’ll go through each x-value one by one, showing you exactly how to calculate the corresponding y-value and form the ordered pair.
1. When x = -2
- Substitute: Replace 'x' in the equation with -2: y = 2(-2) - 5
- Multiply: 2 * -2 = -4, so the equation becomes: y = -4 - 5
- Subtract: -4 - 5 = -9, therefore: y = -9
- Ordered Pair: The ordered pair is (-2, -9).
2. When x = -1
- Substitute: Replace 'x' in the equation with -1: y = 2(-1) - 5
- Multiply: 2 * -1 = -2, so the equation becomes: y = -2 - 5
- Subtract: -2 - 5 = -7, therefore: y = -7
- Ordered Pair: The ordered pair is (-1, -7).
3. When x = 0
- Substitute: Replace 'x' in the equation with 0: y = 2(0) - 5
- Multiply: 2 * 0 = 0, so the equation becomes: y = 0 - 5
- Subtract: 0 - 5 = -5, therefore: y = -5
- Ordered Pair: The ordered pair is (0, -5).
4. When x = 1
- Substitute: Replace 'x' in the equation with 1: y = 2(1) - 5
- Multiply: 2 * 1 = 2, so the equation becomes: y = 2 - 5
- Subtract: 2 - 5 = -3, therefore: y = -3
- Ordered Pair: The ordered pair is (1, -3).
5. When x = 2
- Substitute: Replace 'x' in the equation with 2: y = 2(2) - 5
- Multiply: 2 * 2 = 4, so the equation becomes: y = 4 - 5
- Subtract: 4 - 5 = -1, therefore: y = -1
- Ordered Pair: The ordered pair is (2, -1).
The Completed Table
Now that we've calculated the y-values for each x-value, let's put it all together in our completed table:
| x | y | Ordered Pair |
|---|---|---|
| -2 | -9 | (-2, -9) |
| -1 | -7 | (-1, -7) |
| 0 | -5 | (0, -5) |
| 1 | -3 | (1, -3) |
| 2 | -1 | (2, -1) |
See? It wasn't so bad after all! We've successfully completed the table by substituting each x-value into the equation, solving for y, and writing the ordered pairs. This table gives us a clear picture of how the x and y values relate to each other in the equation y = 2x - 5.
Understanding Ordered Pairs and Graphing
So, what do these ordered pairs actually mean? Each ordered pair represents a specific point on the graph of the line y = 2x - 5. The x-value tells us how far to move horizontally from the origin (0, 0), and the y-value tells us how far to move vertically.
For example, the ordered pair (-2, -9) means we move 2 units to the left of the origin (because x is -2) and 9 units down (because y is -9). Similarly, the ordered pair (2, -1) means we move 2 units to the right and 1 unit down.
If you were to plot these points on a graph and connect them with a straight line, you would get the graph of the equation y = 2x - 5. This is a fantastic way to visualize the relationship between x and y and see how the equation creates a line.
Why is This Important?
You might be wondering,
