Slope and line
All lines can be expressed in the form y = kx + b, where k is the slope of the line and b is a constant. The form of the above straight line is called a solved form.
Slope
The slope shows how steep a line rises or falls. When the slope is positive, the line is ascending and when the slope is negative, the line is descending. The higher the slope, the steeper the line rises. The lower the slope, the steeper the line descends.
Slope
When two points of a line are known, the slope can be calculated with this formula .
Example 1
Find the slope of the line below.
We select the points (1,1) and (3,5) from the line. The difference between the y-coordinates of the points is 4 and the difference between the x-coordinates is 2. The slope is therefore
The lines f, g, h and i
We will find the slope for all these lines
We select two points
The slope of line f is k = 2.
The slope of line i is k = -1
The line h is horizontal, so the difference between the y-coordinates is 0. The slope k = 0
The line g is vertical, so the difference between the x-coordinates is 0. In this case, the divisor would be 0, so, the slope cannot be defined. Lines that are parallel to the y-axis have no slope.
The constant term b of a line
The equation for a line implements the rule between the x and y coordinates. For example, the line y = 2x + 2 tells us that the y-coordinate of a point is obtained by multiplying the x-coordinate by two and adding two to the product. If we want to calculate the point of the line where the x-coordinate is 1, we place x = 1 in the equation for the line. In this case, y = 4 would be obtained, so that point of the line is (1,4).
In the above equation of a line, the constant term is 2. If you want to know at what point on the line x = 0, we substitute it into the equation of the line. We then have y = 2, which is a constant term. So, when x = 0, we are on the y-axis. In other words, the standard term tells us at which point the line intersects the y-axis.
Example 3
Determine the equations for the line in example 2.
Line f intersects the y-axis at the point (0,5), so b = 5 and the equation of the line is y = 2x + 5
Line i intersects the y-axis at a point (0,3), so b = 3 and the equation of the line y = -x + 3
Line h has only a constant term so the equation of the line is y = 5
Line g has no slope and does not intersect the y-axis. The equation of the line is x = 2
Example 4
Find the equation for the line passing through points (2,5) and (4,9).
The line is in the form of y = kx + b
First we determine the slope
The slope is 2, so the line is given the form of y = 2x + b. The equation of the line implements the rule between the coordinates of both points (2,5) and (4,9). Let us substitute a point (2,5) into the line equation.
The constant term is 1, so the equation of the line is
The general form of a line
Like a circle, a line has a normal-shaped equation. All terms are moved to the left of the equation and the equation is modified so that the coefficients of the terms are integers. The coefficient of a term with x is positive and this term is first in the equation.
Example 5
We set the equation of the line below to the normal form
Intersection
The line intersects the y-axis when x = 0 and the x-axis when y = 0.
Example 6
Find the points of intersection of the line y = 2x - 4 and the coordinate axes.
Solution
From the constant term of a line, it is directly seen that the line intersects at the point on the y-axis (0, -4)
Let y = 0 so we get the point of intersection of the x-axis.
2x - 4 = 0, which gives x = 2. The line intersects at the x-axis point (2,0)
Example 7
Find the intersection of the lines y = 3x - 6 and y = 2x - 4
Solution
The point of intersection is the point common to both lines. The equations of the lines give a pair of equations. Mark the lines as equal.
3x - 6 = 2x - 4, the solution of the equation is x = 2 and substituting this in one of the equations of the lines gives y = 0
Lines intersect at a point (2,0)
Perpendicularity of lines
Lines are perpendicular to each other if the slopes of the lines are inverses of each other's opposite numbers. In other words, the product of the slopes is -1
Example 8
Show that the lines below are perpendicular to each other.
We set the equations to slope-intercept form
The slopes are -2 and 1/2, so
The lines are perpendicular to each other
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