x 0 1 2 3 4 5 y 6 6.5 7 7.5 8 8.5
The Correct Answer and Explanation is:
It looks like you’ve provided a set of data points with xxx and yyy values:
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | 6 | 6.5 | 7 | 7.5 | 8 | 8.5 |
It appears that you’re interested in analyzing or solving something related to this data. Since the yyy-values increase consistently as xxx increases, we could be looking at a linear relationship between xxx and yyy. Let’s explore the situation further.
Step 1: Check if the relationship is linear
To confirm if the relationship between xxx and yyy is linear, we can calculate the rate of change in yyy with respect to xxx. This is done by finding the difference between successive yyy-values and checking if it’s constant.
- y(1)−y(0)=6.5−6=0.5y(1) – y(0) = 6.5 – 6 = 0.5y(1)−y(0)=6.5−6=0.5
- y(2)−y(1)=7−6.5=0.5y(2) – y(1) = 7 – 6.5 = 0.5y(2)−y(1)=7−6.5=0.5
- y(3)−y(2)=7.5−7=0.5y(3) – y(2) = 7.5 – 7 = 0.5y(3)−y(2)=7.5−7=0.5
- y(4)−y(3)=8−7.5=0.5y(4) – y(3) = 8 – 7.5 = 0.5y(4)−y(3)=8−7.5=0.5
- y(5)−y(4)=8.5−8=0.5y(5) – y(4) = 8.5 – 8 = 0.5y(5)−y(4)=8.5−8=0.5
Since the difference is constant, this confirms that the relationship is linear.
Step 2: Find the equation of the line
We can use the general equation for a straight line:
y=mx+cy = mx + cy=mx+c
Where:
- mmm is the slope of the line.
- ccc is the y-intercept.
The slope mmm can be calculated using the change in yyy and xxx. Since the change in yyy is 0.5 for every 1 unit change in xxx, the slope is: m=0.5m = 0.5m=0.5
Next, we use any point on the line to find ccc. Using the point (0,6)(0, 6)(0,6): 6=0.5(0)+c⇒c=66 = 0.5(0) + c \quad \Rightarrow \quad c = 66=0.5(0)+c⇒c=6
Thus, the equation of the line is: y=0.5x+6y = 0.5x + 6y=0.5x+6
Step 3: Verify the equation with the data
We can plug the values of xxx into the equation to see if we get the corresponding values of yyy:
- For x=0x = 0x=0, y=0.5(0)+6=6y = 0.5(0) + 6 = 6y=0.5(0)+6=6
- For x=1x = 1x=1, y=0.5(1)+6=6.5y = 0.5(1) + 6 = 6.5y=0.5(1)+6=6.5
- For x=2x = 2x=2, y=0.5(2)+6=7y = 0.5(2) + 6 = 7y=0.5(2)+6=7
- For x=3x = 3x=3, y=0.5(3)+6=7.5y = 0.5(3) + 6 = 7.5y=0.5(3)+6=7.5
- For x=4x = 4x=4, y=0.5(4)+6=8y = 0.5(4) + 6 = 8y=0.5(4)+6=8
- For x=5x = 5x=5, y=0.5(5)+6=8.5y = 0.5(5) + 6 = 8.5y=0.5(5)+6=8.5
Since the calculated values of yyy match the provided data, the equation y=0.5x+6y = 0.5x + 6y=0.5x+6 is correct.
Conclusion
The data represents a linear relationship, and the equation that models this relationship is y=0.5x+6y = 0.5x + 6y=0.5x+6. This means for every 1 unit increase in xxx, yyy increases by 0.5 units. The y-intercept is 6, meaning when x=0x = 0x=0, y=6y = 6y=6.
