Question:
Here are two points: (-3,4), (1,7). What is the slope of the line between them?
Solution:
The slope-intercept form of a line is given by the following equation:
y = mx + b
where m is the slope of the line, and b is the y-coordinate of the y-intercept. Now, By definition, the slope of the line that passes through two points (X1,Y1) and (X2,Y2) is:
[tex]m\text{ = }\frac{Y2-Y1}{X2-X1}[/tex]in our case, we have that:
(X1, Y1) = (-3,4)
(X2,Y2) = (1,7)
Replacing these points in the slope equation we obtain:
[tex]m\text{ = }\frac{Y2-Y1}{X2-X1}=\text{ }\frac{7-4}{1-(-3)}=\text{ }\frac{7-4}{1-(-3)}=\text{ }\frac{3}{1+3}=\text{ }\frac{3}{4}[/tex]then, the equation of the given line is:
[tex]y\text{ = }\frac{3}{4}x+b[/tex]Now, to find b, replace any point (x,y) of the line, in the above equation. For example, take the point (x,y) = (1,7) and replace it on the above equation:
[tex]7\text{= }\frac{3}{4}\cdot1+b[/tex]this is equivalent to:
[tex]7\text{= }\frac{3}{4}+b[/tex]solve for b:
[tex]b\text{ = 7 - }\frac{3}{4}\text{ = }\frac{25}{4}[/tex]thus, we can conclude that the equation of this line is:
[tex]y\text{ = }\frac{3}{4}x+\text{ }\frac{25}{4}[/tex]
And the corresponding graph is:
Then, the correct answer is: A
