Write the equation of a line in slope intercept form given the table of the line.
| x | y |
|---|---|
| 3 | 3 |
| 4 | 5 |
| 5 | 7 |
| 6 | 9 |
| 7 | 11 |
A. [tex]$y=3 x+1$[/tex]
B. [tex]$y=-2 x$[/tex]
C. [tex]$y=3 x-5$[/tex]
D. [tex]$y=2 x-3$[/tex]
Answer (2)
Calculate the slope using two points from the table: m = 4 − 3 5 − 3 = 2 .
Substitute the slope and one point (3, 3) into the slope-intercept form: 3 = 2 ( 3 ) + b .
Solve for the y-intercept: b = 3 − 6 = − 3 .
Write the equation of the line: y = 2 x − 3 , so the answer is y = 2 x − 3 .
Explanation
Understanding the Problem We are given a table of x and y values that represent points on a line. Our goal is to find the equation of this line in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Calculating the Slope First, we need to calculate the slope ( m ) of the line. We can use any two points from the table. Let's use the points (3, 3) and (4, 5). The slope is calculated as the change in y divided by the change in x :
m = x 2 − x 1 y 2 − y 1 = 4 − 3 5 − 3 = 1 2 = 2
Finding the Y-Intercept Now that we have the slope, we can use one of the points and the slope to find the y-intercept ( b ). Let's use the point (3, 3) and the slope m = 2 . We plug these values into the slope-intercept equation: y = m x + b 3 = 2 ( 3 ) + b 3 = 6 + b
Solving for b Now, we solve for b :
b = 3 − 6 = − 3
Writing the Equation Now we have the slope m = 2 and the y-intercept b = − 3 . We can write the equation of the line in slope-intercept form: y = 2 x − 3
Final Answer The equation of the line in slope-intercept form is y = 2 x − 3 . Comparing this to the given options, we see that it matches option d.
Examples
Understanding linear equations is crucial in many real-world scenarios. For example, if you are tracking the cost of a taxi ride, the initial fee is the y-intercept, and the cost per mile is the slope. By knowing these values, you can predict the total cost of any ride. Similarly, in business, linear equations can help model revenue and expenses, allowing companies to forecast profits and make informed decisions. Linear equations are also used in physics to describe motion with constant velocity, where the slope represents the velocity and the y-intercept represents the initial position.
The equation of the line represented by the table is y = 2 x − 3 in slope-intercept form. The slope was calculated as 2 , and the y-intercept was found to be − 3 . Therefore, the correct answer is option D.
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