Question

What is the difference between point-slope form and slope-intercept form ?

Answer 1

Hint:A straight arrangement of points is referred to as a line. In geometry, a line can be defined as a straight one dimensional figure that has no thickness and extends endlessly in both directions. A straight line is one that has no curves. A straight line is a line that stretches to infinity on both sides. We must be clear in mind with the slope point form and slope intercept form of a straight line.

Complete step by step answer:
Point Slope Form: The point-slope form of a straight line is derived from the idea of determining the equation of a line when slope of a line is given along with one point lying on the line. In other words, point slope form helps us in finding the equation of a line when we are provided with the slope of the line and coordinates of one point lying on it.

Equation of a straight line in point slope form: When a line passes from any arbitrary point with coordinates $\left( {{x_1},{y_1}} \right)$ and its slope is given as m, then its equation of line is given as: $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$. Here, \[\left( {{x_1},{y_1}} \right)\] is the specific point on the line and $\left( {x,y} \right)$ is any other general point on the line.

Example: Write the equation of line passing through a point $\left( { - 2,4} \right)$ and its slope is $ - \dfrac{2}{3}$ ?
The general equation of line passing through a point with coordinates \[\left( {{x_1},{y_1}} \right)\] and having slope as m is $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$.
So, we are given the slope of the line as $ - \dfrac{2}{3}$. Coordinates of a point lying on the straight line is $\left( { - 2,4} \right)$.
So, $m = - \dfrac{2}{3}$, \[{x_1} = - 2\] and \[{y_1} = 4\].
Substituting the above values in the point slope form of a straight line, we get,
$ \Rightarrow \left( {y - 4} \right) = \left( { - \dfrac{2}{3}} \right)\left( {x - \left( { - 2} \right)} \right)$
Simplifying the equation, we get,
$ \Rightarrow \left( {y - 4} \right) = \left( { - \dfrac{2}{3}} \right)\left( {x + 2} \right)$
Cross multiplying the terms of the equation, we get,
$ \Rightarrow 3y - 12 = - 2x - 4$
$ \Rightarrow 3y + 2x = 8$
So, the equation of the line is $3y + 2x = 8$.

Slope Intercept Form: The slope-intercept form of a straight line is derived from the idea of determining the equation of a line when slope of a line is given along with the y intercept of the line. In other words, slope intercept form helps us in finding the equation of a line when we are provided with the slope of the line and coordinates of the point where the line crosses the y axis. The slope and y-intercept can easily be defined or read off from this form, allowing one to identify the characteristics of the straight line without having to see its graph.

Equation of the line: When a line is cutting the y-axis at any arbitrary point $\left( {0,c} \right)$ and slope of the line is given as $m$, then its equation of line is given as: $y = mx + c$. Here, $m$ is the slope of the line which measures the steepness of the line with respect to any horizontal line. Example: Write the equation of a line in a slope intercept form whose $y$ intercept is $3$ and its slope is $ - 5$.
The general equation of line in slope-intercept form is $y = mx + c$.
So, we are given the slope of the line as $ - 5$ and the y intercept of a line as $3$.
Substituting the above values in the slope intercept form of a straight line, we get,
$ \Rightarrow y = - 5x + 3$
So, the equation of the line is $y = - 5x + 3$.

Note: All points on the curve are satisfied by the relationship between variables x and y. The general equation of a straight line is $ax + by + c = 0$ where x and y are variables, and a, b and c are constants. Final answer of each question should be represented in the general form of straight line regardless of whichever form is used in solving for the equation of the line.