A system of linear equations is a set of two or more linear equations. To show that equations are part of the same system they're usually written on top of each other with 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 equations often contain more than one unknown variable, and 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:System of equations can be solved both graphically and algebraically, using the substitution method or elimination method.
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 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
Before substitution is possible, one equation must have an isolated variable. Notice that by adding to both sides of Equation (I), can be isolated.
Substitute the rewritten equation from Step into the other equation by substituting the expression equal to in Equation (II). Now, Equation (II) only has one variable,
The sum of two numbers is One of the numbers is two more than three times the other number. Write a system that represents the given relationships. Then, find the numbers using substitution.
We can use the given information to write two equations. First, we must define our variables. Let the first number be and the other We know that the sum of these numbers is Thus, We also know that one of the numbers, let's say is two more than three times the other number, which is then This gives us the equation Together these two equations create the system To solve this system using substitution, we must substitute one equation into the other. Let's substitute into This will allow us to then solve for
A system of inequalities is a set of two or more inequalities that are solved together. The following system of inequalities contains two conditions on the variables and Systems of inequalities are often illustrated graphically in a coordinate plane, where the inequalities define a region.
To graph the inequalities, begin with the boundary lines. The inequality has the boundary line Since the inequality sign is the line should be dashed, and the region below the line is shaded.
Similarly, has the boundary line The inequality sign is so the line is dashed and the region below the line is shaded.
Notice that the individual solution sets overlap for a portion of the graph. This overlapping region is the solution set of the system. The points in this region are all the points that satisfy both inequalities. In this case, this is the purple region.
Lastly, since the boundary lines in their entirety are not part of the solution set, trim them to only show the borders of the overlapping region.
Marco's mother asks him to buy burritos and tacos from the restaurant near their house. She gives him and instructs him to get enough food so that they can feed people. If burritos cost each and tacos cost each, how many of each can he buy?
The purple region represents the solution set that satisfies both inequalities. Since Marco can't buy a negative number of burritos or tacos we're only interested in the positive values of and
Any point in this region corresponds to a combination on burritos and tacos that costs less than and feeds at least people. Let's look at the corners of this region.
The marked points represent minimum and maximum possibilities.
Since we don't know how hungry the guests are, or what their preferences are, Marco should buy both burritos and tacos. Let's choose a point in the middle of the region.
One possibility is that Marco can purchase tacos and burritos. That way, there will probably be enough food, and he'll have money left. Note that even though decimal numbers are a part of the solution set, the answer should be given in whole numbers, assuming you can't buy a part of a taco or burrito.