Algebra is a mathematics part that deals with letters and symbols that are incorporated in the mathematical expression as a representative of the number or a quantity. These unknown values termed as the 'variables' are calculated through various means. One of the common ways is through calculating the slope. The slope reflects how the straight line angles away from the horizontal.
Therefore, it can be said that the slope is nothing but the ratio of the amount by which ‘y’ increases as the ‘x’ increases by a certain amount. Here, ‘x’ and ‘y’ are two inter-dependent variables of an equation. It can also be defined as ‘rise’ divided by the ‘run,’ representing the steepness of a line, or, in other words, how much increase happens in variable ‘y’ as variable ‘x’ increases. Thus, it helps in knowing whether the line is rising or falling. It is said that over the line, at any point, the slope remains constant; that is, it doesn’t change over a line.
The first of the forms of the linear equation is written with the help of slope and y-intercept. The slope-intercept form, thus, is nothing but a linear equation expression. The synonym to the slope is the Gradient of the line. Both are calculated through the same formula. However, both have a difference; while the slope is a scalar, the gradient is a vector quantity.
To expertise Algebraic mathematics, it is important to learn - the slope of line formula, how to find slope from an equation and slope calculation, and the concept of the gradient of a line. These concepts are explained in detail below to help students develop an understanding of the concept of slope as a whole.
Slope intercept formula
The slope-intercept is given by:-
Where ‘m’ stands for ‘move,’ representing the slope of the given line
And, ‘b’ stands for the beginning of the line, representing the ‘y’ intercept or the ‘y’ coordinates of the given line
This formula is often used to express the equation of the line. There are several ways by which a line can be expressed. Each expression might look different, but all represent the same thing or the same line. One such way of writing the equation is “slope-intercept form.” It is commonly used to describe the equation when the equations are represented over a graph.
Two points on the line are taken up in such a situation, and the slope is calculated. The value attained is the value of ‘m’ in the above-written equation. After calculating the slope, the ‘y’ coordinate is being calculated in the form of (0,b); the value thus arrived represents the value of ‘b’ in the above-written equation.
Once all the values are being calculated, they are plugged in the above equation y=mx+b. For Example – If, suppose the value of the slope is calculated as 8 and intercept as 3, then the equation will be
The expression, thus attained by putting up the calculated value, can be verified by picking up the points on the line and putting them into the equation. If the values selected to satisfy the equation, the expression attained is correct, and if not, then the expression has been wrongly formed.
In the case of the vertical line, the equation of the line will be “x=b,” as the vertical line is a straight line, going up and down; in such a situation, the slope becomes undefined. Similarly, for the horizontal line, the equation of the line will be “y=b” as the slope becomes zero in this case.
How to find Y-intercept
The Y-intercept is the point where a line crosses or passes through the y-axis. For instance, it is given that "y= 2x-2", then "-2" represents the y-intercept in the equation. This is because, we know, the basic slope-intercept form is y=mx+b, where ‘b’ represents the y-intercept. Therefore, if we place the values from the example equation into the slope incept form, we will get the y-intercept. In the above example, it turns out to be “-2.”
Or, in case there is no equation but the coordinates or points are given then, the slope and value of ‘b’ have to be calculated from which the equation can be formed. The formed equation thus will give the required Y-intercept. This can also be explained through the example.
Suppose, there are two coordinates (-3,3) and (3,-1)
Step1 – The slope will be “(y2−y1)/(x2−x1)”
Here, (y2−y1) = -1-3=-4
And, (x2−x1) = 3+3=6
m = -4/6= -2/3
Step 2 – intercept equation form is
b = y+2/3x
Thus, y-intercept will be 1
The slope of a line formula
If, two points of the line are given then, the slope of a line can be calculated through the formula “(y2−y1)/(x2−x1)”. The steps followed will be as below-
Step 1 – from the given two points assign one as (x1/y1) and the other as (x2/y2)
Step 2 – put these value in the formula
Step 3 – perform the operation, to calculate the value
This can be better understood with the help of an example,
Suppose points are (15,8) and (10,7)
Step 1 – Suppose the point (10, 7) represents (x1, y1) and (15, 8) as (x2, y2)
Step 2 – Plugging the value in “(y2−y1)/(x2−x1)” , we get , (8-7)/(15-10)
Step 3 – conducting the given operation, gives '1/5'. Therefore, the slope of the line is '1/5'.
If the slope of the line is positive, it means that it rises from left to right and if it is negative that is less than zero, then the line falls from left to right. If the two lines have the same slope then it represents that the two lines parallel each other. And, if the slope of the two lines is opposite or reciprocal then the lines are said to be perpendicular.
How to find slope from an equation
If, you already have an equation then finding the slope is more easy. In such a case, the value has to be directly picked up from the equation. For instance, the equation is, “ y = 3x+5”, then, the slope will be 3 because, as per the rules, the intercept form is “y=mx+b”, where ‘m’ represents the slope. As the example equation is already in the intercept form, the value will be the one that represents the ‘m’ (the Slope).
If, the equation is not in the intercept form and is something like this 2x+5y=5, then, firstly it has to be turned into the slope-intercept form to find the slope.
How will it be converted into slope-intercept form? Let’s follow below basic steps,
Step 1 – Reorganize the equation by Subtracting ‘2x’ from both sides of the equation, such that we get, 5y=5-2x
Step 2 – To turn the equation into slope-intercept form, divide both the sides by 5, such that we get, y = (-2/5)x+1
Step 3 – Now, the equation is in slope-intercept form, therefore the slope will be the value corresponding to “m”, that is, “-2/5”
Slope calculation formula
The slope calculation formula is given by:-
Slope = Rise/Run
Or, Slope point equation: “(y2−y1)/(x2−x1)”
Or, Slope intercept formula: y=mx+b, “m” represents the slope value
The slope calculation formula thus varies from problem to problem. If there is a slope-intercept equation, use the basic principle, just plug in the value and you have the slope value. But, if the equation is in some different form, convert it into the slope-intercept form, plug in the value to find the slope. Or, lastly, if there is a graph or coordinates (or points) are given, use the slope point equation “(y2−y1)/(x2−x1)”, plug in the point values, perform the operation, and solve for the value of the slope.
Let's learn this through an example, suppose, there are two points (– 5, 8) and Q(4, 10), such that
(X1, y1) = (-5,8 )
And, (X2, y2) = (4,10)
Therefore, by slope point formula -
“(y2−y1)/(x2−x1)” = (10-8)/(4-(-5))=2/9=0.22
if instead of the points, there is an equation, such as 2x+4y=8
In such a case, the slope will be calculated as
Step 1 - reorganize the equation - that is, 4y=8-2x
Step 2 - divide the equation by 4, that is, y=2-0.5x
Step 3 - place the values in the slope-intercept formula to get the slope value; that is, the slope-intercept form is y=mx+b, such that 'm' represents slope. Here' is -0.5
The slope of this equation can also find in another way. The equation "2x+4y=8" is in standard form Ax+By= C, and in standard form, the slope is given by '(-A/B).' therefore, the slope will be "-2/4" = "-0.5"
The gradient of a line formula
The gradient is the synonym for ‘slope.’ It is the rate at which the lines rise/falls vertically for each unit across to the right. It is said that the higher the gradient, the steeper it will be. Also, the negative gradient represents the downward sloping line.
The formula for the gradient of a line is given by:-
“(change in y-coordinate)/(change in x-coordinate)”
Thus, if the gradient of a line is to be found, it can be calculated through the following steps.
Step 1 – Choose any two points on the line.
Step 2 – On the graph, draw a right angle triangle with the line as the hypotenuse.
Step 3 – with the help of the scale, note down, triangles vertical length, and horizontal length.
Step 4 – divide the vertical length by the horizontal length.
Step 5 – the value thus attained is the gradient of the line.
Let’s learn the calculation through an example.
Suppose there are two points, P(– 4, 5) and Q(4, 17), such that
(X1, y1) = (-4,5 )
And, (X2, y2) = (4,17)
Therefore, by gradient formula.
“(y2−y1)/(x2−x1)” = (17-5)/(4-(-4))=12/8=1.5
As the gradient here is positive, it means that the line is sloping upward. However, if it had been negative, it would have meant that the line is sloping downward.
The gradient can also be zero, that would mean that the line is a straight line crossing horizontally. And if the calculated gradient is undefined, it means the line is a straight line crossing vertically. Apart from it, the gradient of lines can also tell the relationship each line share with the other. If the gradient of two lines multiplies to give -1, it means the lines are perpendicular.
For instance, there are two equations, "y=3-5x" and "y=(1/5)x". The first equation gradient is '-5', and for the second equation, it is (1/5). Multiplying the gradient of two lines gives "-1", representing that the lines are perpendicular and will be crossing each other at a ninety-degree angle. Similarly to the gradient of the two lines is the same, it will mean the lines are parallel to each other.
For instance, there are two lines,
" y = 2x+5" and "y= 2x-7"
The gradient for " y = 2x+5" will be 2
And, The gradient for " y = 2x-7" will also be 2
As the two lines share the same gradient, the lines are parallel to each other. This can be verified by drawing the two lines on the graph.
In conclusion, the slope is the rate of change with respect to another variable. It helps to define the relationship between the two variables. In the linear equation, the slope can easily be found through the slope-intercept formula. And, If there are just coordinates, the slope can be found through the slope point formula.
Similar to the concept of the slope is the concept of the gradient. The two are much the same with the same formula and meaning. For linear equations, both are calculated the same way.
Knowing the slope concept is important because it is far more broad and applicable as is being thought of. It can represent the acceleration or velocity of the change in various fields.
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