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Linear Equations
A linear equation is one that can be written in the form y = mx + b , where m and b are
real numbers, and y and x are variables. It is linear as it is a polynomial of order 1 in x
(highest exponent of x is 1). Linear equations can also be written in the form
Ax + By = C , which is far more common in solving equations vs the former’s use in
graphing.
Linear equations can be graphed on the Cartesian Plane, where x and y form a
coordinate pair of points (written as (x, y ) ), and, as the name suggests, they create a
straight line when considering all the points that satisfy the linear equation. In the
equation above, b is the ​y-intercept (value of y when x is 0), and m is the ​slope of the
graph (rise over run)​. In the Ax + By = C form, the slope can easily be taken by
rearranging the variables, then dividing by the coefficient B (y =
−A
x
B
+
C
B
).
Let us take a look at the illustration below as an example:
Image from: ​https://www.ipracticemath.com/learn/algebra/algebra_linear_nonlinear_equations
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The line y = 2x + 3 is plotted above, showing several points. Substituting each of these
points back into the equation will indeed show equality, e.g. (1, 5) ⇒ 2(1) + 3 = 5
If we were to determine the line’s equation from its graphical representation, we can first
determine the slope via the equation
y 2 −y 1
x2 −x1
, which is the mathematical equivalent of the
aforementioned rise/run. Substituting points (1, 5) and (− 1, 1) back into the equation,
we get m =
5−1
1−(−1)
= 2 . As for the y-intercept, we simply need to check where the line
intersects the y-axis, or the vertical, bolded line.
Solving linear equations
Linear equations can be solved quite easily as it requires nothing more than basic
algebra. For example, we look at the equation 6(x − 5) = 2(x + 3)
First, we distribute the constants to obtain an equation without parenthetical expressions:
6x − 30 = 2x + 6
Then, we may isolate the variables on one side and the constants on the other to obtain
6x − 2x = 6 + 30
Simplifying, we obtain
4x = 36
and finally,
x=9
Solving for x in linear equations may often be more complex than this cited example, but
do
not
require
more
than
the
above
information.
For
example,
solving
7x + 26 − 4(3x + 5) = 5(x + 2) uses naught more than the steps accompanied with
GEMDAS applications from arithmetic.
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Systems of equations
Sometimes, questions may give two linear equations, looking for a coordinate pair (x,y)
that satisfy both. Graphically, this pair is ​the point of intersection​, as it lies on both lines.
For example, let us use y = 5x + 6 and y = 2x − 4 . We call this pair of equations a ​system​.
To solve for the unknowns in the system, we can proceed by either using ​substitution ​or
elimination.​ For the sake of demonstration, we’ll provide an example for both and show
that they lead to the same answer.
Let our two linear equations be:
X + Y = 80
12Y + 19Y = 1184
Substitution
This involves expressing one variable in terms of the other. Here, the first equation is
simpler as the leading coefficients of X and Y are 1, so we let X = 80 − Y .
Plugging this into the second equation (as we know the values X and Y must be the
same in both equations), we obtain 12(80 − Y ) + 19Y = 1184 . This finally gives Y = 32 ,
which we can plug back into the first equation to obtain X = 48 . We can instead make Y
the subject of the equation Y = 80 − X or even manipulate the second equation to
produce the same results, albeit perhaps slower. This is an exercise left to the reader.
Elimination
In solving by elimination, we aim to get rid of a variable by means of subtracting or
adding the two equations. In this case, we will choose to eliminate X first. As equality is
preserved when manipulating both sides of an equation, what we can do is magnify the
first equation by a factor of -12.
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This gives:
− 12X − 12Y = − 960
12X + 19Y = 1184
By adding these two equations, we obtain 7Y = 224 and Y = 32 . Notice that X is
eliminated in this process, making it much simpler to solve in just one variable. From
here, we can again obtain that X = 48 , and again, we may opt to instead eliminate Y to
produce the same results. Again, this is an exercise left to the reader.
Parallel and Perpendicular lines
Graphically speaking, parallel lines are lines that never intersect. In the coordinate plane,
this can be represented as two lines with equal slopes and different y-intercepts, as the
lines will never “catch up” to one another, having the same value for rise/run at every
increment. For example, y = 2x + 3 and y = 2x − 5 are parallel to one another.
Perpendicular lines, on the other hand, are those whose slopes are negative reciprocals
of each other, i.e. their product is -1. For example, the lines y = 2x + 3 and y = − 12 x + 5
are perpendicular.
Horizontal and Vertical Lines
Recall that the equation for the slope of a line is
y 2 −y 1
x2 −x1
. For example, a horizontal line
passing through the points (0,3) and (1,3). Using this equation, we find that the slope is
0
1
,
which is equal to 0. All horizontal lines have a slope of 0, because when the rise in
rise/run is zero, any number that zero divides is equal to zero,
Vertical lines, on the other hand, have an undefined slope, because they have an infinite
rise over a run of 0, and we know that anything divided by 0 is undefined.
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Collinear Lines
Collinear lines are simply a set of lines that have the same equation when simplified. For
example, y = 2x + 4 and 2y = 4x + 8 are collinear to one another.
Midpoint of Two Points
If one were to connect two points by a straight line, then infinitely many points would also
be present on this line, among which is the ​midpoint. ​Particularly, the midpoint, denoted
as M, of line segment AB, is the ​point on AB such that AM = BM. ​Given points
A(x1 , y 1 ) and B(x2 , y 2 ) , then
M(
x! +x2 y 1 +y 2
​ articularly, the midpoint has coordinates
2 , 2 ). P
equal to the ​average​ of the two endpoints.
Distance Formula
It is known that the shortest path between any two points is a straight line segment. The
length of this segment is easy to calculate had the points been on a vertical or horizontal
line, but this is not always the case. Thus, the distance formula states given two points
A(x1 , y 1 ) and B(x2 , y 2 ) , ​AB =
√(x
1
− x2 )2 + (y 1 − y 2 )2 .
Solving for the equation of a line given different informations
• The standard form​: Ax + By = C
where A, B, and C are all integers, and A, B =/ 0 .
• The slope-intercept form​: y = mx + b
If both slope (m) and y-intercept (b) are known, the above equation can be used. The
equation’s form is extremely popular for graphing as one can directly obtain
corresponding values for y from plugging in x.
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• The point-slope form​: y − y 1 = m(x − x1 )
Given just a single point that lies on a line and the line’s respective slope, this form can
be used to determine the line’s equation. It can then be expanded and rearranged to
obtain the slope-intercept form.
• The two-point form​:
y − y1 =
y 2 −y 1
x2 −x1 (x
− x1 )
This form is similar to the point-slope form, other than m being replaced by the rise/run
expression.
y
• The two-intercept form​: ax + b = 1
Given that a line has x-intercept (a,0) and y-intercept (0,b), the equation above can be
used to calculate the line’s equation without getting the slope. Getting rid of the
denominators will lead to the equation’s standard form.
Example:
Find the equation of the line which passes through the points (1, 2) and (4, 8)
Solution: We will present two solutions. The first involves solving for the equation’s slope
and y-intercept, while the second uses the two-point equation shown above.
Slope-intercept
By the slope formula,
m =
8−2
4−1
= 2.
Plugging this into the slope-intercept form, we have y = 2x + b
By plugging in (1,2) into this equation, we obtain:
2 = 2(1) + b ⇒ b = 0
Thus, the line has equation y = 2x . We can verify that (4,8) satisfies this equation.
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Two-Point
By
letting
y − 2 =
8−2
(x
4−1
(x1 , y 1 ) = (1, 2) and (x2 , y 2 ) = (4, 8) ,
we
have
the
equation
to
be
− 1) ⇒ y − 2 = 2(x − 1) ⇒ y = 2x , which is the same as above.
Intersection of Linear Equations
In regards to the intersection of two linear equations, there are only 3 possible cases.
They are as follows:
Case 1: The 2 lines only intersect at ​1 point. ​In the point slope form, this occurs when the
values for m are different.
Case 2: The lines overlap, and there are ​infinitely many points of intersection. In the
point-slope form, this occurs when the values for m and b are the same.
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Case 3: the lines are parallel, and therefore ​do not have​ any intersection points. In the
point-slope form, this occurs when the value for m is the same, but not b.
Linear Inequalities
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Linear inequalities are expressions that involve inequalities of linear functions. These
inequalities four distinct forms:
Greater than
> or “greater than” signifies that the solution is the set of coordinates ​above ​the linear
function. Keep in mind that the linear function itself is ​not included in the set of
coordinates and is graphed with a ​dotted line.
Graph with a positive slope.
Graph with negative slope
Less than
< or “less than” is signifies that the solution is the set of coordinates ​below ​the linear
function. Keep in mind that the linear function itself is ​not included in the set of
coordinates and is graphed with a ​dotted line
Graph with a positive slope.
Graph with a negative slope
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Greater than or equal to
≥ ​or “greater than or equal to” ​signifies that the solution is the set of coordinates ​above
the linear function. Keep in mind that the linear function itself is included ​in the set of
coordinates and is graphed with a full line
Graph with a positive slope.
Graph with a negative slope
Less than or equal to
≤ ​or “less than or equal to” ​signifies that the solution is the set of coordinates ​below the
linear function. Keep in mind that the linear function itself is included ​in the set of
coordinates and is graphed with a full line
Graph with a positive slope.
Graph with a negative slope