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Here are a few recommended readings before getting started with this lesson.
Here are a few practice exercises before getting started with this lesson.
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.
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.
Two important concepts will be discussed here.
(I): LHS/2=RHS/2
(I): Write as a sum of fractions
(I): ca⋅b=ca⋅b
(I): Put minus sign in front of fraction
(I): Calculate quotient
(I): Identity Property of Multiplication
Now that the equations are both written in slope-intercept form, they can be graphed on the same coordinate plane.
The point where the lines intersect is the solution to the system.
The lines appear to intersect at (1.5,2.5). Therefore, this is the solution to the system — the value of x is 1.5 and the value of y is 2.5.
Tearrik is throwing a party and bought some sodas and chips.
He bought 21 items and spent $51. The cost of a soda is $2 and the cost of a bag of chips is $3. This can be modeled by a system of equations.Start by writing both linear equations in slope-intercept form.
(I): LHS−x=RHS−x
(II): LHS−2x=RHS−2x
(II): LHS/3=RHS/3
(II): Write as a sum of fractions
(II): ca⋅b=ca⋅b
(II): Put minus sign in front of fraction
(II): Calculate quotient
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.
The system of equations in the previous example had one solution. This leads to two important definitions.
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.
(II): LHS−4x=RHS−4x
(II): LHS/(-2)=RHS/(-2)
(II): Write as a difference of fractions
(II): -b-a=ba
(II): ca⋅b=ca⋅b
(II): Put minus sign in front of fraction
(II): a−(-b)=a+b
(II): Calculate quotient
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.
(I): LHS−x=RHS−x
(II): LHS−4x=RHS−4x
(II): LHS/4=RHS/4
(II): Write as a sum of fractions
(II): Put minus sign in front of fraction
(II): ca⋅b=ca⋅b
(II): Calculate quotient
(II): Identity Property of Multiplication
In the last example, one of the systems had infinitely many solutions. Systems of equations with infinitely many solutions have a special name.
Furthermore, systems with no solutions also have a special name.
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.
Mark modeled the number of sedans and trucks sold with a system of equations.Graph both equations on the same coordinate plane.
(II): LHS+4x=RHS+4x
(II): LHS/2=RHS/2
(II): Write as a sum of fractions
(II): ca⋅b=ca⋅b
(II): Calculate quotient
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.
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.
With the content learned in this lesson, the challenge presented at the beginning can be solved. It has been said that 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.
Variable | Meaning of the Variable |
---|---|
x | Number of years since 2015 |
y | Number of students that participate in high school basketball (thousands) |
Variable | Meaning of the Variable |
---|---|
x | Number of years since 2015 |
y | Number of students that participate in high school soccer (thousands) |