Equations of a line

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Any non-vertical line in a plane can be represented by an equation that specifies

the y-intercept, i.e., where does the line cross the y-axis
the slope of the line, i.e., the ratio of the rise of the line to the run of the line,

rise / run
=
change in y / change in x
=
Δ y / Δ x
=
y₂ − y₁ / x₂ − x₁

Below we will look at finding the equation of a line if we are given certain information. These explanations need to be written for use in a number of different courses. For most of those courses we write the slope-intercept form of a linear equation as y = m*x + b, where m is the slope and b is the second coordinate of the y-intercept. In statistics, however, we write the slope-intercept form of a linear equation as y = a + b*x, where a is the second coordinate of the y-intercept and b is the slope. Because we have two different forms of the equation of a line the presentation will be given in a two-column table. The left column will use the y = m*x + b form and the right column, probably of interest only to students in statistics, will use the y = a + b*x form.

Using the slope-intercept form: *y = mx + b**	Using the slope-intercept form: *y = a + bx**
Situation 1: Find the equation of a line if we are told that the slope is ¾ and the y-intercept is at the point (0,5).
We have the y-intercept form as *y = mx + b, where m is the slope and b is the second coordinate of the y-intercept. But we told that m is ¾ and the y-intercept is the point (0,5), which means that b is 5. Therefore, our equation is y = ¾x + 5*	We have the y-intercept form as *y = a + bx, where a is the second coordinate of the y-intercept and b is the slope. But we told that b is ¾ and the y-intercept is the point (0,5), which means that a is 5. Therefore, our equation is y = 5 + ¾x*
Situation 2: Find the equation of a line if we are told that the slope is 2.5 and the the line contains the point (4,7).
We have the y-intercept form as *y = mx + b, where m is the slope and b is the second coordinate of the y-intercept. But we told that m is 2.5. However, we have no idea about the value of the y-intercept. We can at least start writing the equation of the line as y = 2.5x + b* But we know that the point (4,7) has to make that equation work, so we know that *7 = 2.54 + b We can simplify this to 7 = 10 + b and then solve that for b, ⁻3 = b But that means that we now know the value of b so our final equation becomes y = 2.5x + ⁻3*	We have the y-intercept form as *y = a + bx, where a is the second coordinate of the y-intercept and b is the slope. But we told that b is 2.5. However, we have no idea about the value of the y-intercept. We can at least start writing the equation of the line as y = a + 2.5x* But we know that the point (4,7) has to make that equation work, so we know that *7 = a + 2.54 We can simplify this to 7 = a + 10 and then solve that for a, ⁻3 = a But that means that we now know the value of a so our final equation becomes y = ⁻3 + 2.5x*
Situation 3: Find the equation of a line if we are told that y-intercept is (0,2) the line contains the point (6,-7).
We have the y-intercept form as *y = mx + b, where m is the slope and b is the second coordinate of the y-intercept. But we told that b is 2. However, we have no idea about the value of the slope. We can at least start writing the equation of the line as y = mx + 2* But we know that the point (6,-7) has to make that equation work, so we know that *-7 = m6 + 2 We can simplify this to -9 = m6* and then solve that for m, ⁻9/6 = m or ⁻3/2 = m But that means that we now know the value of m so our final equation becomes *y = (⁻3/2)x + 2**	We have the y-intercept form as *y = a + bx, where a is the second coordinate of the y-intercept and b is the slope. But we told that a is 2. However, we have no idea about the value of the slope. We can at least start writing the equation of the line as y = 2 + bx* But we know that the point (6,-7) has to make that equation work, so we know that *-7 = 2 + b6 We can simplify this to -9 = b6* and then solve that for b, ⁻9/6 = b or ⁻3/2 = b But that means that we now know the value of b so our final equation becomes *y = 2 + (-3/2)x**
Situation 4: Find the equation of a line if we are told that the line contains the point (-4,-2) and the point (8,-5).
We have the y-intercept form as *y = mx + b, where m is the slope and b is the second coordinate of the y-intercept. However, we have no idea about the value of the either the slope or the y-intercept. We do know two points on the line. If we say that point 1 is (x₁,y₁) = (8,-5) and that point 2 is (x₂,y₂) = (-4,-2) then we can compute the slope of the line containing those two points as m = y₂ − y₁ / x₂ − x₁ = -5 − (-2) / 8 − (-4) = -3 / 12 = -1 / 4 . Now we know that m=(-1/4). We can at least start writing the equation of the line as y = (-1/4)x + b* But we know that the point (8,-5) has to make that equation work, so we know that *-5 = (-1/4)8 + b We can simplify this to -5 = -2 + b and then solve that for b, ⁻3 = b But that means that we now know the value of b so our final equation becomes y = (⁻1/4)x + ⁻3*	We have the y-intercept form as *y = a + bx, where a is the second coordinate of the y-intercept and b is the slope. However, we have no idea about the value of the either the slope or the y-intercept. We do know two points on the line. If we say that point 1 is (x₁,y₁) = (8,-5) and that point 2 is (x₂,y₂) = (-4,-2) then we can compute the slope of the line containing those two points as b = y₂ − y₁ / x₂ − x₁ = -5 − (-2) / 8 − (-4) = -3 / 12 = -1 / 4 . Now we know that b=(-1/4). y = a + (-1/4)x* But we know that the point (8,-5) has to make that equation work, so we know that *-5 = a + (-1/4)8 We can simplify this to -5 = a + (-2) and then solve that for a, ⁻3 = a But that means that we now know the value of a so our final equation becomes y = -3 + (-1/4)x*

Here are a few more important concepts related to the equation of a line.

Parallel lines have the same slope but different y-intercepts.
Perpendicular lines, other than horizontal and vertical lines, have slopes that multiply to yield -1.
A slope of 0 indicates a horizontal line.
We cannot represent a vertical line in slope intercept form.
Lines with a positive slope go uphill reading left to right. The larger the slope the steeper the hill.
Lines with a negative slope go downhill reading left to right. The more negative the slope the steeper the hill.
If we represent the slope as p / q
then every time x increases by q we must have y increase by p. For this we understand that "increasing" by a negative number is really a decrease.
If we represent the slope as p / q
then if we increase x by 1 then we will increase y by p / q
. For this we understand that "increasing" by a negative number is really a decrease.
We often talk about the dependent and the independent variables in a linear equation. Note that we set up our slope-intercept form of the equation with the y all by itself on one side of the equation. The x is on the other side as part of an expression. This makes it easy for us to just pick any value for x and then use that value to compute the expression which then becomes the required value of y. Because, to do this, it is easy to choose any value of x we call x the independent variable. Because once we have chosen a value for x we compute the value of the expression and that becomes the value of y, that is we have no choice for that y value, we call y the dependent variable.

Using the slope-intercept form: *y = mx + b**	Using the slope-intercept form: *y = a + bx**
Situation 1: Find the equation of a line if we are told that the slope is ¾ and the y-intercept is at the point (0,5).
We have the y-intercept form as *y = mx + b, where m is the slope and b is the second coordinate of the y-intercept. But we told that m is ¾ and the y-intercept is the point (0,5), which means that b is 5. Therefore, our equation is y = ¾x + 5*	We have the y-intercept form as *y = a + bx, where a is the second coordinate of the y-intercept and b is the slope. But we told that b is ¾ and the y-intercept is the point (0,5), which means that a is 5. Therefore, our equation is y = 5 + ¾x*
Situation 2: Find the equation of a line if we are told that the slope is 2.5 and the the line contains the point (4,7).
We have the y-intercept form as *y = mx + b, where m is the slope and b is the second coordinate of the y-intercept. But we told that m is 2.5. However, we have no idea about the value of the y-intercept. We can at least start writing the equation of the line as y = 2.5x + b* But we know that the point (4,7) has to make that equation work, so we know that *7 = 2.54 + b We can simplify this to 7 = 10 + b and then solve that for b, ⁻3 = b But that means that we now know the value of b so our final equation becomes y = 2.5x + ⁻3*	We have the y-intercept form as *y = a + bx, where a is the second coordinate of the y-intercept and b is the slope. But we told that b is 2.5. However, we have no idea about the value of the y-intercept. We can at least start writing the equation of the line as y = a + 2.5x* But we know that the point (4,7) has to make that equation work, so we know that *7 = a + 2.54 We can simplify this to 7 = a + 10 and then solve that for a, ⁻3 = a But that means that we now know the value of a so our final equation becomes y = ⁻3 + 2.5x*
Situation 3: Find the equation of a line if we are told that y-intercept is (0,2) the line contains the point (6,-7).
We have the y-intercept form as *y = mx + b, where m is the slope and b is the second coordinate of the y-intercept. But we told that b is 2. However, we have no idea about the value of the slope. We can at least start writing the equation of the line as y = mx + 2* But we know that the point (6,-7) has to make that equation work, so we know that *-7 = m6 + 2 We can simplify this to -9 = m6* and then solve that for m, ⁻9/6 = m or ⁻3/2 = m But that means that we now know the value of m so our final equation becomes *y = (⁻3/2)x + 2**	We have the y-intercept form as *y = a + bx, where a is the second coordinate of the y-intercept and b is the slope. But we told that a is 2. However, we have no idea about the value of the slope. We can at least start writing the equation of the line as y = 2 + bx* But we know that the point (6,-7) has to make that equation work, so we know that *-7 = 2 + b6 We can simplify this to -9 = b6* and then solve that for b, ⁻9/6 = b or ⁻3/2 = b But that means that we now know the value of b so our final equation becomes *y = 2 + (-3/2)x**
Situation 4: Find the equation of a line if we are told that the line contains the point (-4,-2) and the point (8,-5).
We have the y-intercept form as *y = mx + b, where m is the slope and b is the second coordinate of the y-intercept. However, we have no idea about the value of the either the slope or the y-intercept. We do know two points on the line. If we say that point 1 is (x₁,y₁) = (8,-5) and that point 2 is (x₂,y₂) = (-4,-2) then we can compute the slope of the line containing those two points as m = y₂ − y₁ / x₂ − x₁ = -5 − (-2) / 8 − (-4) = -3 / 12 = -1 / 4 . Now we know that m=(-1/4). We can at least start writing the equation of the line as y = (-1/4)x + b* But we know that the point (8,-5) has to make that equation work, so we know that *-5 = (-1/4)8 + b We can simplify this to -5 = -2 + b and then solve that for b, ⁻3 = b But that means that we now know the value of b so our final equation becomes y = (⁻1/4)x + ⁻3*	We have the y-intercept form as *y = a + bx, where a is the second coordinate of the y-intercept and b is the slope. However, we have no idea about the value of the either the slope or the y-intercept. We do know two points on the line. If we say that point 1 is (x₁,y₁) = (8,-5) and that point 2 is (x₂,y₂) = (-4,-2) then we can compute the slope of the line containing those two points as b = y₂ − y₁ / x₂ − x₁ = -5 − (-2) / 8 − (-4) = -3 / 12 = -1 / 4 . Now we know that b=(-1/4). y = a + (-1/4)x* But we know that the point (8,-5) has to make that equation work, so we know that *-5 = a + (-1/4)8 We can simplify this to -5 = a + (-2) and then solve that for a, ⁻3 = a But that means that we now know the value of a so our final equation becomes y = -3 + (-1/4)x*