How can you find the slope and the y-intercept of a linear equation from a table of values?
The Correct Answer and Explanation is :
To find the slope and the y-intercept of a linear equation from a table of values, follow these steps:
Step 1: Understand the Table
A table of values typically lists pairs of x and y coordinates. For example:
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
Step 2: Identify the Points
From the table, you can identify several coordinate pairs (x, y). For instance, using the first two pairs (1, 2) and (2, 4).
Step 3: Calculate the Slope
The slope (m) of a linear equation is calculated using the formula:
[
m = \frac{y_2 – y_1}{x_2 – x_1}
]
Using the two points from the table:
- Let ( (x_1, y_1) = (1, 2) ) and ( (x_2, y_2) = (2, 4) )
Substituting these values into the formula gives:
[
m = \frac{4 – 2}{2 – 1} = \frac{2}{1} = 2
]
Step 4: Find the Y-Intercept
The y-intercept (b) is the value of y when x = 0. If the table does not directly include x = 0, you can use the slope-intercept form of a linear equation ( y = mx + b ) to solve for b.
Using one of the points, let’s use ( (1, 2) ):
[
2 = 2(1) + b
]
[
2 = 2 + b \implies b = 0
]
Final Equation
Now, you can express the linear equation as:
[
y = 2x + 0
]
Conclusion
In summary, by selecting pairs of points from the table, calculating the slope using the difference in y-values over the difference in x-values, and then determining the y-intercept by substituting known values back into the slope-intercept equation, you can effectively derive the linear equation from the table. The final form ( y = mx + b ) summarizes the relationship between x and y, where ( m ) is the slope and ( b ) is the y-intercept.