12 Skills Refresher: Systems of Linear Equations (5.1 – 5.2) — Intermediate Algebra
5.1 Graphed Solutions — Intermediate Algebra
Often, it is necessary to find the coordinates that are shared by two or more equations. There are multiple methods to find these shared values. While not giving values that are precise, graphing these equations make it possible to see the approximate solution and what type of solution it is.
Problems like 
have been solved in this textbook by adding 4 to both sides and then dividing by 3 (solution is 
). There are also methods to solve equations with more than one variable in them. It turns out that, to solve for more than one variable, it is necessary to have the same number of equations as variables. For example, to solve for two variables such as 
and 
two equations are required. When there are several equations that must be solved, that is called a system of equations. When solving a system of equations, the solution must work in both equations. This solution is usually given as an ordered pair, 
The following example illustrates a solution working in both equations.
Example 5.1.1
Show 
is a solution to the equations 
and 
Using the coordinates 
check to see if 
is a solution for both equations.
The coordinate is a solution for both equations, which means that the lines that they represent will intersect at when they are drawn on a graph.
Note that the graph of an equation yields a picture of all its solutions. When two equations are graphed on the same coordinate plane, this displays not only the solutions of both equations, but where these solutions intersect. For instance, these solutions will show if the linear equations intersect at a point, over a line, or not at all.
In the following examples, each of the three possible types of solutions will be explored.
Example 5.1.2
Identify if any common intersection exists between the following linear equations: 
and 
First, graph 
The slope is 
and the 
-intercept is 3. For the first point, choose the intercept, and choose the second one using the slope. Draw a line through these two points to generate the first line.
Now, graph the second equation, 
, using its slope of 
and intercept of −2.
The lines cross at the coordinate 
To check to see if this intersection is correct, substitute 
and into the two original equations.
The coordinate 
is the shared point between both linear equations. This type of intersection is a unique solution that is called consistent and independent.
It is also possible to have a situation in which the same linear equation is graphed twice. Such an equation is easy to create.
Take the equation 
:
These are all the same equation, and if any two of them were graphed, the result would be the exact same line. This type of intersection has many solutions and is called consistent and dependent.
Example 5.1.3
Find the intersection of the linear equations 
and 
Plot these equations using their intercepts:
For :
when 
when 
For 
:
when
when
The two lines from the previous example have the exact same intercepts and, when graphed, draw the exact same line. Since the two graphs have solutions, it is defined as being consistent. However, the many solutions means they are dependent.
Lines that are parallel will never intersect and, as a result, will never have a solution or a shared intersection. Any system of equations having no solution is defined as being inconsistent. Parallel equations are identical except for having different intercepts. This means that the equation 
is parallel to:
Each of the above equations is parallel and will never intersect with each other.
Example 5.1.4
Find the intersection of the linear equations 
and 
Plot these equations using their intercepts:
For :
when 
when 
For 
:
when 
when 
5.2 Substitution Solutions — Intermediate Algebra
While solving a system by graphing has advantages, it also has several limitations. First, it requires the graph to be perfectly drawn: if the lines are not straight, it may result in the wrong answer. Second, graphing is challenging if the values are really large—over 100, for example—or if the answer is a decimal that the graph will not be able to depict accurately, like 3.2134. For these reasons, graphing is rarely used to solve systems of equations. Commonly, algebraic approaches such as substitution are used instead.
Example 5.2.1
Find the intersection of the equations 
and 
Since 
substitute 
for the in 
The result of this looks like:
Now solve for the variable :
Once the -coordinate is known, the -coordinate is easily found.
To find use the equations 
and 
:
These lines intersect at and 
, or at the coordinate 
This means the intersection is both consistent and independent.
Example 5.2.2
Find the intersection of the equations 
and 
To solve this using substitution, or must be isolated. The first equation is the easiest in which to isolate a variable:
Substituting this value for into the second equation yields:
The equations are identical, and when they are combined, they completely cancel out. This is an example of a consistent and dependent set of equations that has many solutions.
Example 5.2.3
Find the intersection of the equations 
and 
The second equation looks to be the easiest in which to isolate a variable, so:
Substituting this into the first equation yields:
The variables cancel out, resulting in an untrue statement. These are parallel lines that have identical variables but different intercepts. There is no solution, and these are inconsistent equations.
