To find the equation of the line that fits the given data points, we need to determine the slope and y-intercept of the line. Here's the step-by-step process:
1. Identify the given data points:
- (0, 3)
- (1, 5)
- (2, 7)
- (3, 9)
- (4, 11)
- (5, 13)
2. Calculate the change in y-values and x-values between any two points:
- From the points (0, 3) and (1, 5):
- Change in y (Δy): [tex]\( 5 - 3 = 2 \)[/tex]
- Change in x (Δx): [tex]\( 1 - 0 = 1 \)[/tex]
3. Determine the slope (m) of the line:
- Slope [tex]\( m \)[/tex] is given by the ratio of the change in y to the change in x:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{2}{1} = 2
\][/tex]
4. Calculate the y-intercept (b) using the equation of a line [tex]\( y = mx + b \)[/tex]:
- We can use any point to find the y-intercept. Let's use the point (0, 3):
[tex]\[
y = mx + b \implies 3 = 2 \cdot 0 + b
\][/tex]
- Simplifying, we find:
[tex]\[
b = 3
\][/tex]
5. Write the final equation of the line:
- Using the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex], the equation of the line is:
[tex]\[
y = 2x + 3
\][/tex]
Therefore, the complete equation describing how [tex]\( y \)[/tex] changes with [tex]\( x \)[/tex] is:
[tex]\[
y = 2x + 3
\][/tex]
