Respuesta :
The correct answer (though not listed) is:
y = 21.25x + 10.5
Explanation:
First we find the rate of change, or slope, between successive points. The formula for slope is:
[tex] m=\frac{y_2-y_1}{x_2-x_1} [/tex]
Using the coordinates of the first two points, we have:
[tex] m=\frac{95.5-53}{4-2}
\\
\\=\frac{42.5}{2}
\\
\\=21.25 [/tex]
Using this same formula, we can see that the rate of change (slope) remais constant throughout the entire data set, so the set is linear.
Next we use point-slope form to write the equation, then transform it by isolating y. Point-slope form is:
[tex] y-y_1=m(x-x_1) [/tex]
Using our slope, 21.25, and the first point (2, 53), we have:
[tex] y-53=21.25(x-2)
\\y-53=21.25x-42.5 [/tex]
To isolate y, add 53 to each side:
y-53+53 = 21.25x-42.5+53
y = 21.25x+10.5
Answer:
[tex]y = 21.25x+10.5[/tex]
Step-by-step explanation:
Using slope-intercept form:
The equation of line is given by:
[tex]y = mx+b[/tex] .....[1]
where, m is the slope and b is the y-intercept.
Here, y represents the distance from the light house and x represents the number of hours.
As per the statement:
The data table represents the distance between a well-known lighthouse and a cruise ship over time.
From the given table:
Consider any two points:
(2, 53) and (4, 95.5)
Using slope formula:
[tex]\text{Slope} =\frac{y_2-y_1}{x_2-x_1}[/tex]
Substitute the given points we have;
[tex]m = \frac{95.5-53}{4-2} =\frac{42.5}{2} = 21.25[/tex]
Substitute in [1] we have;
[tex]y = 21.25x+b[/tex]
Substitute any points from the given table
Let x = 6 and y = 138
Solve for b:
[tex]138 = 21.25 \cdot 6+b[/tex]
⇒[tex]138 = 127.5+b[/tex]
Subtract 127.5 from both sides we have;
⇒[tex]10.5=b[/tex]
We get, the equation of line:
[tex]y = 21.25x+10.5[/tex]
Therefore, the equation represents the distance (y) from the lighthouse, based on the number of hours (x) is, [tex]y = 21.25x+10.5[/tex]
