Question

Wirte an equation for the line that passes through (1,0) and (3,4).

Answer 1

The equation of a line in slope-intercept form is:
y = mx + b
You need to find two things, m and b
m is the slope, and the slope is the difference in y over the difference in x
Take your two given points (1, 0) and (3, 4)
the difference in y is 4 - 0 = 4
the difference in x is 3 - 1 = 2
m = 4/2 = 2
Now insert one point and m into the equation.
y = mx + b
Let's use (1, 0), and m = 2
0 = 2(1) + b
0 = 2 + b
Subtract 2 from both sides
-2 = b
Since we now know m = 2 and b = -2, we insert those values into the equation of a line:
y = mx + b
y = 2x -2 Final answer

Answer 2

To find the equation of a line that passes through two point, you need to figure out two things:
1) the slope of the line
2) the y-intercept of the line

The slope tells you how many y spaces the line moves every time the x moves one.
So if the slope of the line is 2, then if you increase x by 1, then the y will go UP 2. If the slope is -2, then if you increase x by 1, then the y will go DOWN 2.

Let's call the slope the letter "m". In order to find "m" using two points, you just plug the two points into this equation:
\[m= \frac{ y_2-y_1 }{ x_2-x_1}\]
You already have the first point (1,0), meaning: \[x_1=1\] and \[y_1=0\]
and the second point (3,4), meaning: \[x_2=3\] and \[y_2=4\]

Plug those in to find m
\[m= \frac{ y_2-y_1 }{ x_2-x_1}=\frac{ 4-0 }{ 3-1}=2\]

Now that we have the slope, we need to find the y-intercept. Let's call the y-intercept the letter "b". To find "b", use this equation here:
\[y=mx+b\]
Since we already solved for m, all we're missing from that equation is b, so plug in either one of your points (1,0) or (3,4) into the equation to solve for b. Let's use (1,):
\[y=mx+b\]
\[\rightarrow b=y-mx=(0)-2(1)=-1\]

Now that we have both m and b, leave x and y alone and plug in m and b to find the equation of the line:
\[y=mx+b=2x-2\]
and that's your answer
\[y=2x-2\]