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 by 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
Find the slope for all the 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 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 the form y = kx + b
First we determine the slope
The slope is 2, so the line is given the form 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) in the line equation.
The constant term is 1, so the equation of the line is
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