Question

How do you find the Y-intercept with two points?

Answer 1

Hint: First assume two points, then obtain the slope of the line passing through the points. Then write the slope intercept form of an equation, then substitute the value of the slope and then substitute any one of the points that lies on the line to obtain the value of the y intercept.

Formula used:
The equation of line is \[(y-y_{1})=m (x - x_{1})\]
Where m is the slope of the line and the line passes through the point \[\left(x,y\right)\] and \[\left(x_{1},y_{1}\right)\].

The slope intercept form of an equation is \[y = mx + c\] .
\[m\] denotes the slope of the line.
\[c\] denotes the y-intercept of the line.

The slope of a line passes through the points \[\left(x_{1},y_{1}\right)\] and \[\left(x_{2},y_{2}\right)\] is \[m = \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\].

Complete step-by-step solution:
Assume that a line passes throught the points \[\left(x_{1},y_{1}\right)\] and \[\left(x_{2},y_{2}\right)\].
Hence the slope of the line is \[m = \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\].
Substitute the value of m in the equation \[(y-y_{1})=m (x - x_{1})\]:
 \[(y-y_{1})=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}} (x - x_{1})\]
Simplify the above equation:
\[\Rightarrow (y-y_{1})(x_{2}-x_{1})=({y_{2}-y_{1}}) (x - x_{1})\]
\[\Rightarrow y(x_{2}-x_{1})-y_{1}(x_{2}-x_{1})=x({y_{2}-y_{1}}) - x_{1}({y_{2}-y_{1}})\]
\[\Rightarrow y(x_{2}-x_{1})=x({y_{2}-y_{1}}) - x_{1}({y_{2}-y_{1}})+y_{1}(x_{2}-x_{1})\]
Divide both sides by \[(x_{2}-x_{1})\]:
\[y=x\dfrac{{y_{2}-y_{1}}}{(x_{2}-x_{1})} - x_{1}\dfrac{({y_{2}-y_{1}})}{(x_{2}-x_{1})}+y_{1}\]
Compare the above equation with \[y = mx + c\]:
\[m=\dfrac{{y_{2}-y_{1}}}{(x_{2}-x_{1})} \]
\[c = - x_{1}\dfrac{({y_{2}-y_{1}})}{(x_{2}-x_{1})}+y_{1}\]
We know that \[m=\dfrac{{y_{2}-y_{1}}}{(x_{2}-x_{1})} \]
Substitute \[m=\dfrac{{y_{2}-y_{1}}}{(x_{2}-x_{1})} \] in \[c = - x_{1}\dfrac{({y_{2}-y_{1}})}{(x_{2}-x_{1})}+y_{1}\]:
\[c = - mx_{1}+y_{1}\]
The y- intercept of line that passes through two points is, \[-x_{1}\dfrac{({y_{2}-y_{1}})}{(x_{2}-x_{1})}+y_{1}\] or \[ - mx_{1}+y_{1}\]

Note: The y-intercept of a line is a point where the line cuts the y-axis. If we have the equation of line, then calculate the y-intercept by puting \[ x = 0\] in the equation. The coordinates of y-intercept is in the form \[(0,a)\].