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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'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. {x+y=3(I)xy=1(II) \begin{cases}x+y=3 & \, \text {(I)}\\ x-y=1 & \text {(II)}\end{cases} 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 x=2x=2 and y=1.y=1. These coordinates make the sides equal in both equations. The solution is usually written as a point: (2,1). (2,1).

System of equations can be solved both graphically and algebraically, 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, {2y=-2x+8x=y1,\begin{cases}2y=\text{-} 2x+8 \\ x=y-1, \end{cases} 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.y. For the first equation, divide both sides by 2:2\text{:} 2y=-2x+8y=-x+4. 2y=\text{-} 2x+8 \quad \Leftrightarrow \quad y=\text{-} x+4. The second equation can be rewritten by adding 11 on both sides and rearranging it.
x=y1x=y-1
x+1=yx+1=y
y=x+1y=x+1
The second equation, written in slope-intercept form, is y=x+1,y=x+1, meaning that the system can be written as {y=-x+4y=x+1. \begin{cases}y=\text{-} x+4 \\ y=x+1. \end{cases}

2

Graph the lines

Next, graph the lines on the same coordinate plane. Here, the yy-intercepts are b1=4b_1=4 and b2=1b_2=1 the slopes are m1=-1m_1=\text{-} 1 and m2=1.m_2=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).(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.
Exercise

In a football game, the home team, the Mortal Wombats, defeated the Fearless Seagulls by 1313 points. The total score for both teams was 41.41. What was the final score?

Solution

To begin, we'll use variables to represent the different quantities. Let ww be the number of points the Wombats scored and ss be the number of points the Seagulls scored. The Wombats scored 1313 more points than the Seagulls. Thus, the difference between ww and ss can be written as w=s+13. w=s+13. The total amount of points was 41,41, so the sum of ww and ss is w+s=41. w+s=41. Both of these equations must be true simultaneously, giving us the following system of equations. {w=s+13w+s=41 \begin{cases}w=s+13 \\ w+s=41 \end{cases} We can solve the system by graphing. First, let's write the second equation in slope-intercept form by subtracting ss on both sides. {w=s+13w=-s+41 \begin{cases}w=s+13 \\ w=\text{-} s+41 \end{cases} Now, we can graph the lines. Since the scores cannot be negative, we only graph the lines for positive values of ss and w.w.

Now, we can identify the point of intersection.

The point of intersection is (14,27).(14,27). This means, the Wombats scored 2727 points and the Seagulls scored 14.14.

info Show solution Show solution
Method

Substitution Method

The substitution method is an algebraic means of finding the solution(s) to a system of equations by substituting a variable's equivalent value into another equation in the system. {y4=2x(I)9x+6=3y(II)\begin{cases}y-4=2x & \, \text {(I)}\\ 9x+6=3y & \text {(II)}\end{cases} The general idea for this example system is to isolate yy in Equation (I) and substitute its equivalent value into Equation (II). By doing so, yy in Equation (II) will be eliminated and it will be possible to solve for x.x.

1

Solve one equation for one variable

Before substitution is possible, one equation must have an isolated variable. Notice that by adding 44 to both sides of Equation (I), yy can be isolated. {y4=2xy=2x+49x+6=3y\begin{cases}y-4=2x \quad \Rightarrow \quad y=2x+4 \\ 9x+6=3y \end{cases}

2

Substitute the variable's equivalent expression

Substitute the rewritten equation from Step 11 into the other equation by substituting the expression equal to yy in Equation (II). {y=2x+49x+6=3(2x+4)\begin{cases}y={\color{#0000FF}{2x+4}} \\ 9x+6=3({\color{#0000FF}{2x+4}}) \end{cases} Now, Equation (II) only has one variable, x.x.

3

Solve the resulting equation

Since the resulting equation from Step 22 only contains one variable, it can be solved using inverse operations.

9x+6=3(2x+4)9x+6=3(2x+4)
9x+6=6x+129x+6=6x +12
3x+6=123x+6=12
3x=63x=6
x=2x=2
The xx-coordinate of the solution to the system is x=2.x=2.

4

Substitute found variable value into either given equation
The solution found in Step 33 can be used to find the other variable. To do this, substitute the solution into either equation and solve. Here, x=2x=2 is substituted into Equation (I).
y=2x+4y=2x+4
y=2(2)+4y=2({\color{#0000FF}{2}})+4
y=4+4y=4+4
y=8y=8
The yy-coordinate of the solution is y=8.y=8. Thus, the solution to the system is (2,8).(2,8).
It's possible to employ this method for systems that include any combination of linear equations, quadratic equations, or inequalites, among others.
Exercise

The sum of two numbers is 17.17. 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.

Solution

We can use the given information to write two equations. First, we must define our variables. Let the first number be xx and the other y.y. We know that the sum of these numbers is 17.17. Thus, x+y=17. x+y=17. We also know that one of the numbers, let's say x,x, is two more than three times the other number, which is then y.y. This gives us the equation x=3y+2. x=3y+2. Together these two equations create the system {x+y=17x=3y+2. \begin{cases}x+y=17 \\ x=3y+2. \end{cases} To solve this system using substitution, we must substitute one equation into the other. Let's substitute x=3y+2x=3y+2 into x+y=17.x+y=17. This will allow us to then solve for y.y.

x+y=17x+y=17
3y+2+y=17{\color{#0000FF}{3y+2}}+y=17
4y+2=174y+2=17
4y=154y=15
y=154y=\dfrac{15}{4}
y=3.75y=3.75
We've found that the yy-coordinate of the solution is y=3.75.y=3.75. We can substitute this value into either equation to solve for the corresponding xx-value. We'll choose the second equation.
x=3y+2x=3y+2
x=33.75+2x=3\cdot {\color{#0000FF}{3.75}}+2
x=11.25+2x=11.25+2
AddTerms
x=13.25x=13.25
Since x=13.25,x=13.25, the solution to the system is (13.25,3.75).(13.25, 3.75).
info Show solution Show solution
Concept

System of Inequalities

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 xx and y.y. {y-0.5x+3y>x \begin{cases}y \leq \text{-} 0.5x+3 \\ y > x \end{cases} Systems of inequalities are often illustrated graphically in a coordinate plane, where the inequalities define a region.

Method

Graphing a System of Linear Inequalities

Individual inequalities can be interpreted graphically as the area above or below a boundary line. When all inequalitites in a system are graphed, the solution to the system is the overlapping region of the individual solution sets. For example, the following system can be solved in this way. {x+y<7x+2y<10 \begin{cases}x+y<7 \\ x+2y<10 \end{cases}

1

Write the inequalities in slope-intercept form
To be able to graph the inequalities individually, start by writing them in slope-intercept form.
{x+y<7(I)x+2y<10(II)\begin{cases}x+y<7 & \, \text {(I)}\\ x+2y<10 & \text {(II)}\end{cases}
{y<-x+72y<-x+10\begin{cases}y<\text{-} x + 7 \\ 2y<\text{-} x + 10 \end{cases}
{y<-x+7y<-0.5x+5\begin{cases}y<\text{-} x + 7 \\ y<\text{-} 0.5x + 5 \end{cases}

2

Graph the inequalities in a coordinate plane

To graph the inequalities, begin with the boundary lines. The inequality y<-x+7y<\text{-} x + 7 has the boundary line y=-x+7.y=\text{-} x + 7. Since the inequality sign is <,<, the line should be dashed, and the region below the line is shaded.

Similarly, y<-0.5x+5y<\text{-} 0.5 x + 5 has the boundary line y=-0.5x+5.y=\text{-}0.5x+5. The inequality sign is <,<, so the line is dashed and the region below the line is shaded.

3

Find the overlapping region

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.

Exercise

Marco's mother asks him to buy burritos and tacos from the restaurant near their house. She gives him $90 \$90 and instructs him to get enough food so that they can feed 1010 people. If burritos cost $5 \$5 each and tacos cost $3 \$ 3 each, how many of each can he buy?

Solution
There are a number of conditions in this exercise. Let's start with making sense of them. Let's use bb to represent the number of burritos that Marco buys, and tt to represent the number of tacos. Since the burritos cost $5\$ 5 each, the total amount spent on just burriots is 5b.5b. Similarly, since a taco costs $3,\$ 3, the total cost of tacos is 3t.3t. Therefore, The total cost can be expressed as 5b+3t. 5b+3t. Since Marco has $90,\$ 90, he cannot spend more than this. The total must be less than or equal to 90.90. That means, 5b+3t90. 5b+3t\leq 90. We also know that Marco needs to buy enough food to feed 1010 people. Assuming no one wants to share a taco or a burrito, the total amount of dishes must be at least 10.10. This gives b+t10. b+t\geq 10. Since both of these inequalities must hold true, we get the following system of inequalities. {5b+3t90b+t10 \begin{cases}5b+3t\leq 90 \\ b+t\geq 10 \end{cases} 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.b.
5b+3t905b+3t\leq 90
Solve for bb
5b-3t+905b\leq \text{-} 3t+90
b-3t5+905b\leq \text{-} \dfrac{3t}{5} + \dfrac{90}{5}
b-0.6t+18b\leq \text{-} 0.6t+18
The first inequality can be expressed as b-0.6t+18.b\leq \text{-} 0.6t+18. Writing the second inequality in slope-intercept form can be done in one step. Specifically, by subtracting tt on both sides. b+t10b-t+10 b+t \geq 10 \quad \Leftrightarrow \quad b \geq \text{-} t+10 The system can now be expressed as {b-0.6t+18b-t+10.\begin{cases}b\leq \text{-} 0.6t+18 \\ b \geq \text{-} t+10. \end{cases} 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 bb and t.t.

Any point in this region corresponds to a combination on burritos and tacos that costs less than $90\$ 90 and feeds at least 1010 people. Let's look at the corners of this region.

The marked points represent minimum and maximum possibilities.

  • (10,0)(10,0) and (0,10)(0,10) represent Marco buying either 1010 tacos or 1010 burritos. Then he'd feed exactly 1010 people, and have money remaining.
  • (0,18)(0,18) and (30,0)(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 1818 burritos or 3030 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 1212 tacos and 88 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.

info Show solution Show solution
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