Section 6.1 Solutions to a System of Equations
Definition 6.1.
A solution to an equation is a pointExample 6.2.
Let's see if
Indeed,
Checkpoint 6.3.
Determine if each of the following is a solution to the equation
is not a solution is a solution is not a solution
Determine if each of the following is a solution to the equation
-
In order to determine if this is a solution, we plug in andOf course,
so we ended up with something that is not true. Therefore, is not a solution to this equation. -
In order to determine if this is a solution, we plug in andOf course,
so we ended up with something that is true. Therefore, is a solution to this equation. Remember that a solution is a point, which includes two numbers. This is just a single number, so it cannot be a solution to the equation.
Definition 6.4.
A system of equations is a set of more than one equation that we want to look at all at the same time.Definition 6.5.
A solution to a system of equations is a point that is a solution to all of the equations in the system at the same time.Example 6.6.
Let's determine if
Just as we did before, we will plug in
Now, let's check the second one:
Since the point only worked for the first equation, but not the second, it is not a solution to this system of equations.
Checkpoint 6.7.
Suppose we have the system of equations:
Check if each of the following is a solution to the system:
The point
is not a solutionThe point
is a solutionThe point
is not a solution
-
To check if
is a solution, we plug it into both equations. Let's start with the first equation:That worked! Now, let's check the second equation:
That one did not work. Since the point has to work for both equations in order to be a solution to the system, the point
is not a solution to this system. -
To check if
is a solution, we plug it into both equations. Let's start with the first equation:That worked! Now, let's check the second equation:
That one also worked! Since it worked for both equations, the point
is a solution to this system. -
To check if
is a solution, we plug it into both equations. Let's start with the first equation:That did not work. At this point, we could plug it into the second equation, but that would end up just being a waste of time. It doesn't matter if it works for the second one if we already know it doesn't work for the first one. It has to work for both in order to be a solution, so the point
is not a solution to this system.