To show that equations are part of the same system, they are usually written on top of each other and have a curly bracket to the left. It's not unusual to add Roman numerals, to be able to refer to the equations individually. Systems of linear equations often contain more than one unknown variable where the solution is the set of coordinates that make all equations true simultaneously. In the example above, the solution is and These coordinates make the sides equal in both equations. The solution is usually written as a point:A system of linear equations can be solved both graphically and algebraically.
To solve a system of linear equations graphically means graphing the lines and identifying the point of intersection.
For example, the following system, can be solved by graphing.
The point where the lines intersect is the solution to the system.
The lines appear to intersect at Thus, this is the solution to the system.
In the coordinate plane, two lines are graphed.
Use the graph to solve the system
We can find the solution to the system by identifying the point of intersection.
In a football game, the home team, the Mortal Wombats, defeated the Fearless Seagulls by points. The total score for both teams was What was the final score?
To begin, we'll use variables to represent the different quantities. Let be the number of points the Wombats scored and be the number of points the Seagulls scored. The Wombats scored more points than the Seagulls. Thus, the difference between and can be written as The total amount of points was so the sum of and is Both of these equations must be true simultaneously, giving us the following system of equations. We can solve the system by graphing. First, let's write the second equation in slope-intercept form by subtracting on both sides. Now, we can graph the lines. Since the scores cannot be negative, we only graph the lines for positive values of and
Now, we can identify the point of intersection.
The point of intersection is This means, the Wombats scored points and the Seagulls scored