Solving Graphically
For systems of equations, we will be dealing with two variables, x and y, and two equations. We can solve the systems of equations by graphing. One method is to make an x/y chart to find and plot the points on the graph.
This table below is the table for the equation x + y = 5. This table gives us the points to plot on the graph for this part of the system.
The other method used to graph and solve systems of equations is by putting both equaitons into slope-intercept form. Once they are in slope-intercept form, then they are graphed.
x + y = 5
x - x + y = 5 - x
y = 5 + (-x)
y = -x + 5
To find the solution when you graph systems, no matter which method you use to graph, you look for the point of intersection. The point of intersection is the solution because that is when there is a point that is the same for both lines. On the graph this point is where the two lines intersect.
When the solution is not exact, for example the equations are equivalent (same slope and y-intercept) or there is no solutions (same slope, different y-intercepts), then they are graphed differently. When the equations are the same, there are infinite solutions. On the graph, it looks like one line, but two lines are overlapping. When there is no solution, the equations will create lines that are parallel, and they will never intersect.
This method of solving the systems of equations by graphing works well when the solution is an integer. When it isn't an integer, you would have to use the graph to estimate the solution. So this method of graphing will not always be exact.
Subtract x from both sides
Add a negative
Commutative Property
Systems of Inequalities
When solving a system of inequalities, the best way to do it is by doing it graphincally. To graph, you put the inequality in slope-intercept form and graph regularly. Although, if it is strictly greater than or less than it is to be a dotted line. If it has "or equal to" then it is a solid line. Also, if it is less than/less than or equal to, shade below the line. If it is greater than/greater than or equal to, shade above the line. In a systems of equations there are two lines. When you shade and graph the lines accordingly, there should be a spot where it is double shaded. This is the solution area, so any point within the solution area works for the inequality.
To check if it is correct, you can pick a point in the solution in area, and enter the coordinates in the inequality.
Another thing to remember is set notation. Set notation is a way to display a solution for and inequality. It looks like {x:x>3}. The number and inequality sign vary depending on what the systems of inequalities is.
