The solution to this system of equations is (-1, 1).
Step-by-step explanation:
A system of equations is a "a finite set of equations for which common solutions are sought". Or, they are a pair (or more) of equations that either intersect, are parallel, or the same line. For the sake of simplicity, we are going to focus on systems of equations that intersect and only have one solution.
A solution to a system of equations that intersect is represented by a point (x, y). This makes sense because at the intersection of two points, there lies a point.
So how do we find this point? We do this by solving the system of equations. We show that there exists a point in which two lines can be satisfied simultaneously. But how can we do that? Trial and error?! Well, no. There are some ways of approaching this.
The first method is graphing. By either using an external source or doing it yourself, graphing two lines and showing where they intersect is one of the more understandable ways of solving a system of equations. The point where they cross is the solution of the system.
The second method is elimination. This is observing that a system of equations can be combined together to eliminate a variable (x or y) and solving for the other in a one-variable equation. For example, say that I have this system of equations:
x - y = 1
x + y = 1
By simply adding the two together, I get:
2x = 2
x = 1.
Since we already have x = 1, we can substitute it back into one of the original equations to get y.
1 + y = 1
y = 0.
So, the solution to this system of equations would be (1, 0). Speaking of substituting ...
The third method is substitution. This is observing that a variable (x or y) can be written in terms of the other and then using that to substitute in and make one of the equations one-variable.
y = 2x + 3
x - y = -2
We can see that y is already in terms of x in the first equation, so let's substitute y for 2x + 3 in the second equation.
x - (2x + 3) = -2
- x - 3 = -2
- x = 1
x = -1
We can substitute x back into one of the original equations.
y = 2(-1) + 3
y = -2 + 3
y = 1
Therefore, the solution to this system of equations is (-1, 1).
