(9, 3), (-6, 7y), m = 3
This question aims to find unknown points from two points and slopes. A two-point form can express the equation of a straight line in a coordinate plane. Equation of a line can be found by various methods depending on the information available. The two-point form is one of the methods. This is used to find the equation of a line when two points lying on the line are given. Some other important forms to represent the equation of a line are slope-intercept form, intercept form, point-slope form, etc.
The two-point form is one of the important forms used to represent a straight line algebraically. The equation of a line represents every point on the line, i.e., it is satisfied by every point on the line. The two-point line form is used to find the equation of a line given two points $(x1, y1)$ and $(x2,y2)$.
Equation of a line in the form of two-points form:
Two-point form of a line passing through these two points is given by:
\[y-y_{1}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})\]
Where $(x,y)$ are variables and $(x_{1},y_{1}) \:and (x_{2},y_{2})$ are points on the line.
A line passing through two points will have an equation of the form. The equation using two points can also be written as:
\[y=mx+c\]
We can find the slope value $m$, the gradient of the line, by creating a right triangle using the coordinates of the two given points . We can then find the value of $c$, the point of intersection $y$, by substituting the coordinates of one point into the equation. The final output can be checked by substituting the coordinates of the second point into the equation.
Expert Answer
Formula for the slope of the line, given two points on that line is given by:
\[m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
Plug the values of the points on the line and the value of the slope to find the value of unknown $y$.
\[3=\dfrac{7y-3}{-6-9}\]
\[3=\dfrac{7y-3}{-15}\]
Cross multiplying and solving for unknown.
\[-45=7y-3\]
\[7y=-42\]
\[y=-6\]
The value of the unknown $y$ is $-6$.
Numerical Result
The value of the unknown $y$ for the two points and the slope is $-6$.
Example
Determine the value of x or y so that the line passing through the given points has the given slope.
(6, 2), (-6, 2y), m = 5
Solution
Formula for the slope of the line, given two points on that line is given by:
\[m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
Plug the values of the points on the line and the value of the slope to find the value of unknown $y$.
\[5=\dfrac{2y-2}{-6-6}\]
\[5=\dfrac{2y-2}{-12}\]
Cross multiplying and solving for unknown.
\[-60=2y-2\]
\[2y=-58\]
\[y=-29\]
The value of the unknown $y$ is $-29$.
The value of the unknown $y$ for the two points and the slope is $-29$.