Solving Systems of Equations By Graphing and Substitution
A system of equations is a collection of equations with the same variables. An example would be:
The solution to a system of 2 equations (if there is one) will always be an ordered pair. It will look like (x,y), the x being where the point coordinates to the x-axis and the y being where the point coordinates to the y-axis.
Solve By Graphing:
Solving by graphing tells you whether there is a solution or not. There are three possible ways that the graph can look:
When the graph has intersecting lines, there is only one solution that will solve the system of equations.When the graph has coinciding lines, there is an infinite number of solutions because both lines graphed are the same. When the graph has parallel lines, there is no solutions that can solve the system of equations.
A system of equations can be classified using these words:
Consistent- when a system has at least one solution
Independent- when a system has exactly one solution
Dependent- when a system has an infinite number of solutions
Inconsistent- when a system has no solutions
There are two forms of an equations you can graph. The first one is y=mx+b. An example would be y=3x+4. The m, or in this case, the 3 represents the slope and the b, or in this case, the 4 represents where the line crosses the y-axis. To graph this form of an equation, you first graph the b. The b is always in the form of (0,b). So, for this example, you would graph (0,4). Next, we do the slope. The slope would be 3/1. On your graph, you start from the point (0,4) and move up 3 on the y-axis. Then, since the bottom number of the fraction is 1, you would move to the right 1 on the x-axis. But, if the slope happened to be negative, you would either move down the y-axis 3 and to the right on the x-axis 1 or up the y-axis 3 and to the left on the y-axis 1 since the negative sign can be put on either the top on the bottom of the fraction.
The second form of and equation you can graph is Ax+By=C form. An example would be 6x+4y=12. To graph this, you need to solve the equation for x and then for y. When you solve for x you act as if the By part doesn’t exist and vice versa. So, solving for the x in this equation would look like this:
6x=12
/6 /6
x=2
Solving this equation for y would look like this:
4y=12
/4 /4
y=3
This means you would graph (2,0) and (0,3). When you graph these 2 points and connect them, then you have your line.
When you graph an equation, the answer is the point where the two lines you graph intersect.
Using what I have told you so far, you can tell that these are intersecting lines, which means there is only 1 solution. That would make this system of equations consistent and independent. By looking at the point of intersection, you can see that the answer is (-2,1).
Solve By Substitution:
Example System of Equations:
2x+y=3
3x-2y=8
To solve using substitution, you first start by solving one of the equations for either x or y. In this situation, solving 2x+y=3 for y will be easiest because y has a coefficient of 1. Solving it for y will look like this:
2x+y=3
-2x -2x
y=3-2x
Next, you sub the equation you solved for (in this example) y and sub it into y in the second equation and solve. It will look like this:
3x-2y=8
3x-2(3-2x)=8
3x-6+4x=8
7x-6=8
+6 +6
7x=14
/7 /7
x= 2
Then, you substitute x=2 into either original equation to find your y. So, in this example, you could either do:
2x+y=3
2(2)+y=3
4+y=3
-4 -4
y= -1
OR
3x-2y=8
3(2)-2y=8
6-2y=8
-6 -6
-2y=2
/-2 /-2
y= -1
and you still get the same answer for y. So, you answer for this equation would be (2,-1).
This is how you solve a system of equations by graphing or substitution!!!