Point of Intersection Formula | Point Gradient & Point Slope Form Formula

Point of Intersection Formula

Have you heard of point of intersection concept in mathematics? If no, don’t panic. Here, we will discuss the point of intersection in detail and how to calculate it either graphically or algebraically. Also, the formula is applicable to a variety of areas like businesses, finance, study, construction, or physics etc.

Have you ever noticed the traffic signal on a road? This is the example of point of intersection that will appear at the point when two roads are meeting up at a point. In mathematics, point of intersection is the point where two lines or curves generally meet.The value of two curves would be same significantly and it can be used at multiple places.

Take another example, if we wanted to represent the revenue of a Company against the costs then point of intersection would define the situation where revenue and costs are significantly the same. Most of the times, this is the breakeven point for a Company. The point can be calculated either graphically or algebraically.

Draw the graph of two equations and see where they will intersect visually. This is not a tough job but can be completed quickly with a deep understanding and practice. In most of the examples, you could analyze that graph is the best technique to find the point of intersection with accuracy.

Sometimes, there are the situation when this is not possible to find the point of intersection graphically then how can you solve the equation. The answer is you can do it algebraically. Solve the equations find the values of x coordinated that would point of intersection for both the equations.

\[\ (x,y) = \frac{b_{1}c_{2}-b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}, \frac{a_{2}c_{1}-a_{1}c_{2}}{a_{1}b_{2}-a_{2}b_{1}}\]

Point Gradient Formula

For a line, the ratio of vertical change to the horizontal change is defined through a point i.e. named as the point of gradient or we can name it as the derivative as well.  In brief, the gradient of a line will be rise divided by the run – rise/run. If m is the gradient point across a line then point gradient formula in mathematics could be given as –

\[\large Point\;Gradient =\frac{y-y_{1}}{x-x_{1}}\]

Point Slope Form Formula

The other popular format for straight line equations is point slope formula. For this purpose, you need to find out the values (x1, y1) and a slope m. Further, plug the values into the formula –

\[\large y-y_{1}=m(x-x_{1})\]

m is the slope of the line.
x1 is the co-ordinates of x-axis.
y1 is the co-ordinates of y-axis.

Don’t scare of subscripts but they are just intended to indicate the points given to you. If you have the generic values for x and y coordinates then it can be directly plugged into the formula to calculate the final output. If you will calculate the values calculated from the slope-intercept form and the point slope form then they are exactly the same.

So, this is your choice which method are you planning to use and which technique suits you the most. Practice the technique and apply it as per your convenience for next mathematics problem.