Solving Mathematical problems not only enhances your skill in this regard, but, it can be a matter of real fun and enjoyment, once, you develop a habit of solving such problems on a regular basis. One such interesting area is that of the Point slope form of any line.
It is for sure, once you will understand the formula in this regard, and start practicing such problems on a regular basis, you will certainly start loving it. What you simply need is rigorous practice and after that, you will develop the expertise to solve the most complex problems based on this formula.
An excerpt of the Form for Point Slope
You will find the formula in this regard,expressed as Y-Y1 =M(X-X1). In this context, M stands for the slope of the line that passes through the points X1 and Y1. While working on this formula, you should substitute the values for the points X1 and Y1 only, while the points X and Y don’t need such substitution.
This formula comes highly helpful, in instance, you are aware of the Slope that a line produces as well as the point through which the line passes. When you substitute the values for the Coordinate Points X1 and Y1, you can derive alternative forms of this equation like the slope intercept.
Assume, the values for X1 and Y1 are 6 and -3 respectively, while the slope of the line is ½. As per the formula, one can express it like Y-6= ½ (X+3).
Is it possible to derive the Point Slope form using 2 points?
Digging deeper into the concept, it is obvious that you will come across a question that if identical computations can be made using 2 points. Likewise, there can be an instance, wherein you are not served with the value of the slope, but you will need to derive the form for Point slope.
In such instances, you need to identify 2 points through which the given line passes, rather than trying to solve the problem with just a single point. In such instances, you need to use the following formula to derive the desired result:
M= (Y2-Y1)/ (X2-X1)
Once, you derive the slope using this formula, you will need to substitute the formula for the Point Slope Form with one of these points. Now, for the choice of the point, you can opt for any one as per your choices, as there are no guidelines about the selection of the point for solving the puzzle.
Throwing more lights on the formula
The basic equation the applies in this regard is, Y-Y1=M(X-X1). Wherein M represents the line slope, and X1 and Y1 are the coordinates for the given line. Now, as the formula of the slope, calculated with the use of 2 points is M= Y2-Y1/X2-X1, it is possible to cross multiply and doing so, one would derive the formula, M(X2-X1) = Y2-Y1.
In such instances, one can replace the points X2 and Y2 with X and y respectively. This will redesign the formula as M(X-X1)= Y-Y1. As this formula complies with the Point Slope form, it is possible to rewrite the formula as Y-Y1 = M(X-X1). This will successfully derive the form for Point Slope that can be utilized for further computations.
How to find the equation for the slope intercept of a line, using the form for Point slope?
After getting the previous insight on the topic, you are likely to wonder, if it is possible to derive the slope intercept for a given line, applying the form for a point slope. Well, it is possible to compute like that, and you will not face hassles in computing in that manner.
Now, let’s substitute M. X1 and Y1 with the values 2, 3, and, -1. Then, following the basic equation Y-Y1= M(X-X1), one derives the form Y+1 =2(X-3) which is the equation for the slope intercept.
Now, if 2 is distributed within the parenthesis for solving the equation, one derive to the following stands:
Y+1 = 2X-6
Subsequently, if Y is isolated, the new equation will look like y+1-1 =2X -6-1, eventually coming to the format, Y= 2X-7 which is the final derivation for the slope intercept, using the form of the line slope of a line.
Overview of some common applications of the formula
If a linear equation is expressed in the point slope format, it is possible to find the slopes, for the corresponding lines as well as the point through which the line passes. This will aid in graphing the point. As for the application of the concept and the formula, it will depend on the intended purpose of use as well as the information provided.
As it is easy to express the point slope format a line the purpose will be served, if you are served with values for the slope and the point through which the line passes.
Rather than plugging the information into the format of the slope intercept, it makes better senses to straightway plug the information into the format for the Line slope that will derive an acceptable formula. This will make it easier to solve the given problem and more importantly, will result in more authentic and acceptable formula.
Alternatively, if you are served with the information for the slope as well as the Y-Intercept, it will be better If you are opting for the format for the slope intercept in such instances.
As usual with any other topics from mathematics, the only way to acquire a mastery on this chapter in regular and rigorous practice. As a matter of tips, you should try to understand the logic between the equation that will enable you to acquire better understanding of it, and subsequently, it will turn easier for you to solve these problems.
This will enable you to attain better mastery in solving algebraic and geometric problems and consolidate your standing as a master in Mathematics. You will surely find the concept of Point Slope Form all the more interesting, as you spend more and more time with it.