In this Grade 9 Saxon Algebra 1 lesson, students learn to solve and classify special systems of linear equations as consistent and independent, consistent and dependent, or inconsistent. Using substitution and slope-intercept form, students identify whether a system has exactly one solution, infinitely many solutions, or no solution, and connect each case to the geometric relationship between the graphed lines. The lesson covers key vocabulary including inconsistent equations, dependent equations, and independent systems within Chapter 7 on Rational Expressions and Radicals.
Section 1
📘 Solving and Classifying Special Systems of Linear Equations
A consistent system will have at least one common solution.
Next, you'll use algebraic methods to solve systems and discover whether they have one, infinite, or zero solutions.
Section 2
Inconsistent Equations
If no common solution exists, the system consists of inconsistent equations. The graphs of inconsistent equations are parallel lines that never intersect, so there is no solution.
Think of two runners on parallel tracks who will never meet. When you solve the system, the variables disappear, leaving a false statement like . This impossible result is your clue that the lines are parallel and will never, ever cross paths. There is no solution, and the system is inconsistent.
Section 3
Example Card: Solving an Inconsistent System
What happens when a system of equations leads to an impossible conclusion? Let's find out with our first key idea, an inconsistent system.
Example Problem
Solve the system: and .
Section 4
Dependent Systems
Dependent systems have an infinite number of solutions. The equations are called dependent equations, and they have identical solution sets because their graphs are the same line.
It's like having two secret identities that are actually the same person! When you solve these systems, the variables vanish and you're left with a true statement, like . This means the equations are identical and describe the exact same line, so any point on that line is a valid solution.
Section 5
Example Card: Solving a Dependent System
Sometimes, two different-looking equations are actually the same in disguise. Let's see how this works with our second key idea, a dependent system.
Example Problem
Solve the system: and .
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Section 1
📘 Solving and Classifying Special Systems of Linear Equations
A consistent system will have at least one common solution.
Next, you'll use algebraic methods to solve systems and discover whether they have one, infinite, or zero solutions.
Section 2
Inconsistent Equations
If no common solution exists, the system consists of inconsistent equations. The graphs of inconsistent equations are parallel lines that never intersect, so there is no solution.
Think of two runners on parallel tracks who will never meet. When you solve the system, the variables disappear, leaving a false statement like . This impossible result is your clue that the lines are parallel and will never, ever cross paths. There is no solution, and the system is inconsistent.
Section 3
Example Card: Solving an Inconsistent System
What happens when a system of equations leads to an impossible conclusion? Let's find out with our first key idea, an inconsistent system.
Example Problem
Solve the system: and .
Section 4
Dependent Systems
Dependent systems have an infinite number of solutions. The equations are called dependent equations, and they have identical solution sets because their graphs are the same line.
It's like having two secret identities that are actually the same person! When you solve these systems, the variables vanish and you're left with a true statement, like . This means the equations are identical and describe the exact same line, so any point on that line is a valid solution.
Section 5
Example Card: Solving a Dependent System
Sometimes, two different-looking equations are actually the same in disguise. Let's see how this works with our second key idea, a dependent system.
Example Problem
Solve the system: and .
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter