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Linear Relationships

Solving Linear Systems

Concept

System of Linear Equations

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 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 x=2 and y=1. 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 by using the substitution method or elimination method.

Method

Solving a System of Linear Equations Graphically

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.

1

Write the equations in slope-intercept form
Start by writing the equations in slope-intercept form by solving for y. For the first equation, divide both sides by 2:
The second equation can be rewritten by adding 1 on both sides and rearranging it.
x=y1
x+1=y
y=x+1
The second equation, written in slope-intercept form, is y=x+1, meaning that the system can be written as

2

Graph the lines

Next, graph the lines on the same coordinate plane. Here, the y-intercepts are b1=4 and b2=1 the slopes are m1=-1 and m2=1.

3

Identify the point of intersection

The point where the lines intersect is the solution to the system.

The lines appear to intersect at (1.5,2.5). Thus, this is the solution to the system.

Note that whenever a system of equations is solved graphically, the solution is approximate.
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Exercise

In a football game, the home team, the Mortal Wombats, defeated the Fearless Seagulls by 13 points. The total score for both teams was 41. What was the final score?

Show Solution
Solution
To begin, we'll use variables to represent the different quantities. Let w be the number of points the Wombats scored and s be the number of points the Seagulls scored. The Wombats scored 13 more points than the Seagulls. Thus, the difference between w and s can be written as
w=s+13.
The total amount of points was 41, so the sum of w and s is
w+s=41.
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 s on both sides.
Now, we can graph the lines. Since the scores cannot be negative, we only graph the lines for positive values of s and w.

Now, we can identify the point of intersection.

The point of intersection is (14,27). This means, the Wombats scored 27 points and the Seagulls scored 14.

Method

Substitution Method

This is an algebraic method of finding the solutions to a system of equations. It consists of substituting an equivalent expression for a variable in one of the equations of the system. Consider an example system of linear equations.
To solve the system by using the Substitution Method, there are four steps to follow.

1

Isolate One Variable in Any of the Equations
The first step is to isolate any variable in any of the equations. For simplicity, in this case the y-variable will be isolated in Equation (I).

2

Substitute the Expression
In the equation where the variable was not isolated, substitute the obtained expression for the variable. In this case, 2x+4 will be substituted for y in Equation (II).
Now, Equation (II) only has one variable, which is x.

3

Solve the Equation With One Variable
Solve the equation that contains only one variable. In this case, Equation (II) will be solved for x.
The value of the x-variable, in this case, is 2.

4

Substitute the Value of the Variable in the Other Equation
Since the value of one variable is known, it can be substituted in the equation that has not been considered yet. In this case, x=2 will be substituted in Equation (I).
Evaluate right-hand side
The value of the y-variable, in this case, is 8. Therefore, the solution to the system of equations, which is the point of intersection of the lines, is (2,8) or x=2, y=8.
If at any step of the method a true statement is obtained, then the lines represented by the equations of the system are coincidental. Therefore the system has infinitely many solutions. Conversely, if at any step a false statement is obtained, then the lines are parallel. In this case, the system has no solution.
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Exercise

The sum of two numbers is 17. 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.

Show Solution
Solution
We can use the given information to write two equations. First, we must define our variables. Let the first number be x and the other y. We know that the sum of these numbers is 17. Thus,
x+y=17.
We also know that one of the numbers, let's say x, is two more than three times the other number, which is then y. This gives us the equation
x=3y+2.
Together these two equations create the system
To solve this system using substitution, we must substitute one equation into the other. Let's substitute x=3y+2 into x+y=17. This will allow us to then solve for y.
x+y=17
3y+2+y=17
4y+2=17
4y=15
y=3.75
We've found that the y-coordinate of the solution is y=3.75. We can substitute this value into either equation to solve for the corresponding x-value. We'll choose the second equation.
x=3y+2
x=33.75+2
x=11.25+2

AddTerms

x=13.25
Since x=13.25, the solution to the system is (13.25,3.75).

Concept

System of Inequalities

A system of inequalities is a set of two or more inequalities involving the same variables. For example, consider the set formed by the following two inequalities.
The solution set of a system of inequalities is the set of all ordered points that satisfy all the inequalities in the system simultaneously. Usually, systems of inequalities are solved by graphing each inequality on the same coordinate plane. By doing so, the entire coordinate plane is divided into different regions. By moving P, explore each region formed by the previous system.
Two lines graphed and four regions
Of the regions formed when a system of inequalities is graphed, the overlapping region represents the solution set of the system.

Method

Solving a System of Linear Inequalities Graphically

A system of linear inequalities can be solved graphically by graphing all inequalities on the same coordinate plane and then finding the region of intersection, if any. For example, consider the following system.
To solve the previous system graphically, these three steps can be followed.

1

Write the Inequalities in Slope-Intercept Form
To be able to graph the inequalities individually, start by writing them in slope-intercept form.

2

Graph the Inequalities

To graph the inequalities, begin with the boundary lines. The inequality y<-x+7 has the boundary line y=-x+7. Since the inequality is strict, the line should be dashed and, in this case, the shaded region lies below the line.

Similarly, y<-0.5x+5 has the boundary line y=-0.5x+5. Since the inequality is not strict, the line is solid. In addition, the region to be shaded lies below the line. This inequality will be graphed on the same coordinate plane.

3

Find the Overlapping Region

Notice that the individual solution sets overlap in a portion of the plane. This overlapping region is the solution set of the system. All the points in this region satisfy both inequalities simultaneously. In the next graph, only the common region is plotted.

Finally, since the boundary lines in their entirety are not part of the solution set, crop them only to show the edges of the overlapping region.

Keep in mind that if there is no overlapping region, the system has no solution.
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Exercise

Marco's mother asks him to buy burritos and tacos from the restaurant near their house. She gives him $90 and instructs him to get enough food so that they can feed 10 people. If burritos cost $5 each and tacos cost $3 each, how many of each can he buy?

Show Solution
Solution
There are a number of conditions in this exercise. Let's start with making sense of them. Let's use b to represent the number of burritos that Marco buys, and t to represent the number of tacos. Since the burritos cost $5 each, the total amount spent on just burriots is 5b. Similarly, since a taco costs $3, the total cost of tacos is 3t. Therefore, The total cost can be expressed as
5b+3t.
Since Marco has $90, he cannot spend more than this. The total must be less than or equal to 90. That means,
5b+3t90.
We also know that Marco needs to buy enough food to feed 10 people. Assuming no one wants to share a taco or a burrito, the total amount of dishes must be at least 10. This gives
b+t10.
Since both of these inequalities must hold true, we get the following system of inequalities.
Solving this system gives all of the possible combinations of burritos and tacos Marco can purchase while buying enough food and staying within his budget. To solve the system, we must graph it. Let's first write the inequalities in slope-intercept form by isolating b.
5b+3t90
Solve for b
5b-3t+90
b-0.6t+18
The first inequality can be expressed as b-0.6t+18. Writing the second inequality in slope-intercept form can be done in one step. Specifically, by subtracting t on both sides.
The system can now be expressed as
We'll graph each inequality by showing its boundary line and shading the appropriate region.

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 b and t.

Any point in this region corresponds to a combination on burritos and tacos that costs less than $90 and feeds at least 10 people. Let's look at the corners of this region.

The marked points represent minimum and maximum possibilities.

  • (10,0) and (0,10) represent Marco buying either 10 tacos or 10 burritos. Then he'd feed exactly 10 people, and have money remaining.
  • (0,18) and (30,0) tell how many of each dish Marco can buy if he used all money on and only bought tacos or burritos. Thus, he can buy 18 burritos or 30 tacos.

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 12 tacos and 8 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.

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