Vocabulary
System of linear equations
Consistent and independent
Consistent and dependent
Inconsistent
Consistent and independent
Consistent and dependent
Inconsistent
What is a System of Linear Equations?
A system of linear equations is two or more linear equations that share an ordered pair solution.
For instance, the system of linear equations graphed below:
For instance, the system of linear equations graphed below:
Line 1: y = -2x - 6
Line 2: y = 3x - 1
This system of linear equations has the solution (-1, -4), because this is the point they have in common.
Line 2: y = 3x - 1
This system of linear equations has the solution (-1, -4), because this is the point they have in common.
Classifying Systems of Linear Equations
See the table below to learn how systems of linear equations are classified:
Consistent and independent systems of linear equations are two (or more) lines that intersect at exactly one point. The solution to this system is the ordered pair (x, y) where the lines intersect.
Consistent and dependent systems of linear equations are two (or more) lines that lie on top of each other. Since these lines lie on top of each other, they technically "cross" infinitely many times. Therefore, this type of system has infinitely many solutions.
Inconsistent systems of linear equations are two (or more) lines that are parallel. Since these lines are parallel, they will never cross; therefore, the there is no solution to the system.
Look at the slideshow to see examples of how different systems are classified and their solutions.
Consistent and dependent systems of linear equations are two (or more) lines that lie on top of each other. Since these lines lie on top of each other, they technically "cross" infinitely many times. Therefore, this type of system has infinitely many solutions.
Inconsistent systems of linear equations are two (or more) lines that are parallel. Since these lines are parallel, they will never cross; therefore, the there is no solution to the system.
Look at the slideshow to see examples of how different systems are classified and their solutions.
Applications
Systems of linear equations are used to compare two different rates of change, two different linear equations, and different data sets. To see an example of this, look at the following scenario:
Joe and Josh each want to buy a game from the Steam store. Joe has $14 and saves $10 a week. Josh has $26 and saves $7 a week. In how many weeks will they have the same amount?
Let's define two variables. Let's let t = total amount of money saved and w = number of weeks.
Joe's equation: t = 10w + 14
Josh's equation: t = 7w + 26
If we graph this system of linear equations, we can see how Joe and Josh's money saving will increase over time:
Joe and Josh each want to buy a game from the Steam store. Joe has $14 and saves $10 a week. Josh has $26 and saves $7 a week. In how many weeks will they have the same amount?
Let's define two variables. Let's let t = total amount of money saved and w = number of weeks.
Joe's equation: t = 10w + 14
Josh's equation: t = 7w + 26
If we graph this system of linear equations, we can see how Joe and Josh's money saving will increase over time:
The number of weeks is the independent variable, so it goes on the x-axis. The amount of money saved given the amount of weeks passed is the y-axis. By examining the graph, we can see that the two lines intersect at the point (4, 54). This means that both Joe and Josh will have the same amount of money, $54, after 4 weeks of saving.
Summary
A system of linear equations is two or more linear equations that share an ordered pair solution.
Systems can be classified in one of three different ways:
1) Consistent and independent (having one solution, and ordered pair);
2) Consistent and dependent (having infinitely many solutions);
3) Inconsistent (having no solutions).
Systems can be classified in one of three different ways:
1) Consistent and independent (having one solution, and ordered pair);
2) Consistent and dependent (having infinitely many solutions);
3) Inconsistent (having no solutions).
Missouri Learning Standards
A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Mathematical Practices
(3) Construct viable arguments and critique the reasoning of others.
(4) Look for and express regularity in repeated reasoning.
(4) Look for and express regularity in repeated reasoning.