Answer:
A linear system of two equations with two variables is any system that can be written in the form.
a
x
+
b
y
=
p
c
x
+
d
y
=
q
ax+by=pcx+dy=q
where any of the constants can be zero with the exception that each equation must have at least one variable in it.
Also, the system is called linear if the variables are only to the first power, are only in the numerator and there are no products of variables in any of the equations.
Here is an example of a system with numbers.
3
x
−
y
=
7
2
x
+
3
y
=
1
3x−y=72x+3y=1
Before we discuss how to solve systems we should first talk about just what a solution to a system of equations is. A solution to a system of equations is a value of
x
x
and a value of
y
y
that, when substituted into the equations, satisfies both equations at the same time.
For the example above
x
=
2
x=2
and
y
=
−
1
y=−1
is a solution to the system. This is easy enough to check.
3
(
2
)
−
(
−
1
)
=
7
2
(
2
)
+
3
(
−
1
)
=
1
3(2)−(−1)=72(2)+3(−1)=1
So, sure enough that pair of numbers is a solution to the system. Do not worry about how we got these values. This will be the very first system that we solve when we get into examples.
Note that it is important that the pair of numbers satisfy both equations. For instance,
x
=
1
x=1
and
y
=
−
4
y=−4
will satisfy the first equation, but not the second and so isn’t a solution to the system. Likewise,
x
=
−
1
x=−1
and
y
=
1
y=1
will satisfy the second equation but not the first and so can’t be a solution to the system.
Now, just what does a solution to a system of two equations represent? Well if you think about it both of the equations in the system are lines. So, let’s graph them and see what we get.
As you can see the solution to the system is the coordinates of the point where the two lines intersect. So, when solving linear systems with two variables we are really asking where the two lines will intersect.
We will be looking at two methods for solving systems in this section.
The first method is called the method of substitution. In this method we will solve one of the equations for one of the variables and substitute this into the other equation. This will yield one equation with one variable that we can solve. Once this is solved we substitute this value back into one of the equations to find the value of the remaining variable.
In words this method is not always very clear. Let’s work a couple of examples to see how this method works.
Example 1 Solve each of the following systems.
3
x
−
y
=
7
2
x
+
3
y
=
1
3x−y=72x+3y=1
5
x
+
4
y
=
1
3
x
−
6
y
=
2
5x+4y=13x−6y=2
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a
3
x
−
y
=
7
2
x
+
3
y
=
1
3x−y=72x+3y=1
Show Solution
b
5
x
+
4
y
=
1
3
x
−
6
y
=
2
5x+4y=13x−6y=2
Show Solution
As with single equations we could always go back and check this solution by plugging it into both equations and making sure that it does satisfy both equations. Note as well that we really would need to plug into both equations. It is quite possible that a mistake could result in a pair of numbers that would satisfy one of the equations but not the other one.
Let’s now move into the next method for solving systems of equations. As we saw in the last part of the previous example the method of substitution will often force us to deal with fractions, which adds to the likelihood of mistakes. This second method will not have this problem. Well, that’s not completely true. If fractions are going to show up they will only show up in the final step and they will only show up if the solution contains fractions.
This second method is called the method of elimination. In this method we multiply one or both of the equations by appropriate numbers (i.e. multiply every term in the equation by the number) so that one of the variables will have the same coefficient with opposite signs. Then next step is to add the two equations together. Because one of the variables had the same coefficient with opposite signs it will be eliminated when we add the two equations. The result will be a single equation that we can solve for one of the variables. Once this is done substitute this answer back into one of the original equations
Hope it helps (◕ᴗ◕)
