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Challenge
Basketball and Soccer
The number of students in the US participating in high school basketball and soccer has steadily increased over the past few years.
The following table shows some information about this.
| High School Sport | Number of Students Participating in 2015 (Thousands) |
Average Rate of Increase (Thousands per Year) |
|---|---|---|
| Basketball | 700 | 20 |
| Soccer | 400 | 40 |
Consider the above table to answer the following questions.
a Write two equations to represent the situation.
b Graph the equations from Part A in the same coordinate plane. Use the graph to predict the approximate year when the number of students participating in these two sports will be the same.
Explore
Lines on the Same Plane
In the graph below, four lines and their corresponding linear equations can be seen on a coordinate plane. Investigate which lines intersect at one point, which lines intersect at infinitely many points, and which lines do not intersect at all.
Discussion
Solving a System of Linear Equations Graphically
Example
Modeling With a System of Linear Equations
Tearrik is throwing a party and bought some sodas and chips.
{x+y=212x+3y=51(I)(II)
Here, x is the number of sodas and y the number of bags of chips Tearrik bought. Solve the system by graphing and find how many sodas and bags of chips he bought. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Sodas <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>:","formTextAfter":null,"answer":{"text":["12"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Chips <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>:","formTextAfter":null,"answer":{"text":["9"]}}
Hint
Start by writing both linear equations in slope-intercept form.
Solution
To start, each equation in the system will be written in slope-intercept form. This means that the variable y will be isolated in both equations.
Now, the slope and the y-intercept of each line will be used to draw the graphs on the same coordinate plane. Since the number of items cannot be negative, only the first quadrant will be considered for the graph. 
{x+y=212x+3y=51(I)(II)
(I), (II): Write in slope-intercept form
SubEqn
(I): LHS−x=RHS−x
{y=-x+212x+3y=51
SubEqn
(II): LHS−2x=RHS−2x
{y=-x+213y=-2x+51
DivEqn
(II): LHS/3=RHS/3
{y=-x+21y=3-2x+51
WriteSumFrac
(II): Write as a sum of fractions
{y=-x+21y=3-2x+351
MovePartNumRight
(II): ca⋅b=ca⋅b
{y=-x+21y=3-2x+351
MoveNegNumToFrac
(II): Put minus sign in front of fraction
{y=-x+21y=-32x+351
CalcQuot
(II): Calculate quotient
{y=-x+21y=-32x+17
Finally, the point of intersection P can be identified.
The point of intersection of the lines is P(12,9). In the context of the situation, this means that Tearrik bought 12 sodas and 9 bags of chips.
Pop Quiz
Finding the Solution to a System of Equations by Graphing
Discussion
Consistent and Independent Systems
The system of equations in the previous example had one solution. This leads to two important definitions.
Concept
Consistent System
A system of equations that has one or more solutions is called a consistent system. For example, consider the following linear systems.
In the graph, it can be seen that the first system has exactly one solution. The second system has infinitely many solutions. Therefore, since both systems have one or more solutions, they are consistent systems.
Example System I{y=x+1y=2x−2Example System II{y=x−12y=2x−2
To determine the number of solutions, each system can be graphed on a coordinate plane.
Concept
Independent System
An independent system is a system of equations with exactly one solution. Consider the following linear system.
Since the system has exactly one solution, it is an independent system.
{y=x+1y=2x−2
To state the number of solutions, both lines of the system can be graphed on the same coordinate plane. Example
Finding the Solution and Classifying a System of Equations
At Tearrik's party, Ignacio found one of Tearrik's homework assignments. He does not want to do Tearrik's homework for him, but Ignacio decides to quiz himself using the assignment.
Help Ignacio determine whether the given systems of equations are consistent systems and whether they are independent systems.
a {y=-2x+64x−2y=-20
{"type":"choice","form":{"alts":["The system is consistent and independent.","The system is neither consistent nor independent.","The system is consistent but not independent.","The system is independent but not consistent."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
b {x+y=34x+4y=12
{"type":"choice","form":{"alts":["The system is consistent and independent.","The system is neither consistent nor independent.","The system is consistent but not independent.","The system is independent but not consistent."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}
Hint
a Solve the system by graphing and determine the number of solutions.
b Solve the system by graphing and determine the number of solutions.
Solution
a The system will be solved graphically. To do so, both linear equations must to be expressed in slope-intercept form. In the first equation, the variable y is already isolated. Therefore, only the second equation needs to be rewritten.
{y=-2x+64x−2y=-20(I)(II)
(II): Write in slope-intercept form
SubEqn
(II): LHS−4x=RHS−4x
{y=-2x+6-2y=-4x−20
DivEqn
(II): LHS/(-2)=RHS/(-2)
{y=-2x+6y=-2-4x−20
WriteDiffFrac
(II): Write as a difference of fractions
{y=-2x+6y=-2-4x−-220
DivNegNeg
(II): -b-a=ba
{y=-2x+6y=24x−-220
MovePartNumRight
(II): ca⋅b=ca⋅b
{y=-2x+6y=24x−-220
MoveNegDenomToFrac
(II): Put minus sign in front of fraction
{y=-2x+6y=24x−(-220)
SubNeg
(II): a−(-b)=a+b
{y=-2x+6y=24x+220
CalcQuot
(II): Calculate quotient
{y=-2x+6y=2x+10
The lines intersect at exactly one point.
Since the system has one or more solutions, it is a consistent system. Furthermore, since it has exactly one solution, it is also an independent system.
b Again, the system of equations will be solved graphically. To do so, both equations will be rewritten in slope-intercept form.
Note that both equations are simplified to the same equation. It can be graphed by using its slope and its y-intercept. 
The lines overlap each other. Therefore, they intersect at infinitely many points. Since the system has one or more solutions, it is a consistent system. However, since it does not have exactly one solution, it is not an independent system.
{x+y=34x+4y=12(I)(II)
(I), (II): Write in slope-intercept form
SubEqn
(I): LHS−x=RHS−x
{y=-x+34x+4y=12
SubEqn
(II): LHS−4x=RHS−4x
{y=-x+34y=-4x+12
DivEqn
(II): LHS/4=RHS/4
{y=-x+3y=4-4x+12
WriteSumFrac
(II): Write as a sum of fractions
{y=-x+3y=4-4x+412
MoveNegNumToFrac
(II): Put minus sign in front of fraction
{y=-x+3y=-44x+412
MovePartNumRight
(II): ca⋅b=ca⋅b
{y=-x+3y=-44x+412
CalcQuot
(II): Calculate quotient
{y=-x+3y=-1x+3
IdPropMult
(II): \IdPropMult
{y=-x+3y=-x+3
Discussion
Dependent and Inconsistent Systems
In the last example, one of the systems had infinitely many solutions. Systems of equations with infinitely many solutions have a special name.
Concept
Dependent System
A dependent system is a system of equations with infinitely many solutions. Consider the following linear system.
The lines overlap each other. Therefore, the system has infinitely many solutions. This means that this is a dependent system.
{2y=4x−4y=2x−2
To find the number of solutions, both lines of the system can be graphed on the same coordinate plane. Furthermore, systems with no solutions also have a special name.
Concept
Inconsistent System
A system of equations that has no solutions is called an inconsistent system. For example, consider the following linear system.
The lines do not intersect. The system has no solution and is therefore an inconsistent system.
{y=2x−1y=2x+2
To determine the number of solutions, the system can be graphed on a coordinate plane. Example
Finding the Solution and Classifying a System of Equations
Mark cannot go to Tearrik's party because he has recently started working with his father at a car dealership. At the dealership, Mark's father sells sedans and trucks.
{y=2x+22y−4x=10(I)(II)
Here, x represents the number of days since Mark started working in the agency. In Equation (I) the variable y represents the number of sedans sold. Similarly, in Equation (II) y represents the number of trucks sold. Mark wants to interpret this system of equations in terms of consistency and independence. Help him do this! {"type":"multichoice","form":{"alts":["The system is a consistent system.","The system is an inconsistent system.","The system is a dependent system.","The system is an independent system."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":[1]}
Hint
Graph both equations on the same coordinate plane.
Solution
To interpret the system in terms of consistency and independence, both linear equations will be drawn on the same coordinate plane. To do so, the slope-intercept form will be used. Since the first equation is already written in this form, the y-variable will be isolated in the second equation.
Now that both equations are written in slope-intercept form, their slopes and y-intercepts can be used to draw the lines. 
{y=2x+22y−4x=10(I)(II)
(II): Write in slope-intercept form
AddEqn
(II): LHS+4x=RHS+4x
{y=2x+22y=4x+10
DivEqn
(II): LHS/2=RHS/2
{y=2x+2y=24x+10
WriteSumFrac
(II): Write as a sum of fractions
{y=2x+2y=24x+210
MovePartNumRight
(II): ca⋅b=ca⋅b
{y=2x+2y=24x+210
CalcQuot
(II): Calculate quotient
{y=2x+2y=2x+5
Next, recall the definitions of consistent, inconsistent, dependent, and independent systems.
| Consistent System | A system of equations that has one or more solutions. |
|---|---|
| Independent System | A system of equations with exactly one solution. |
| Dependent System | A system of equations with infinitely many solutions. |
| Inconsistent System | A system of equations that has no solution. |
With this information in mind, the lines will be considered one more time.
Since the lines do not intersect each other, the system has no solution. Therefore, it is an inconsistent system.
Pop Quiz
Classifying Systems
A system of equations can be consistent or inconsistent. In addition, a consistent system can be independent or dependent. Classify the following systems of equations in terms of consistency and independence.
