Mathematics
Grade 8
15 min
Classify a system of equations by graphing
Classify a system of equations by graphing
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1
Introduction & Learning Objectives
Learning Objectives
Accurately graph two linear equations on the same coordinate plane.
Identify the intersection point of two lines, if one exists, from their graph.
Visually determine if two lines are parallel or coincident.
Classify a system of equations as consistent or inconsistent based on its graph.
Classify a consistent system as independent or dependent based on its graph.
State the number of solutions (one, none, or infinitely many) for a system of equations by interpreting its graph.
Have you ever wondered if two different paths will ever cross, or if they'll always stay side-by-side? 🗺️ In math, we can use graphs to find out how two equations relate to each other!
In this lesson, you'll learn how to graph two linear equations together and use their vis...
2
Key Concepts & Vocabulary
TermDefinitionExample
System of Linear EquationsTwo or more linear equations considered together. We are looking for values that satisfy all equations simultaneously.The equations $y = 2x + 1$ and $y = -x + 4$ form a system of linear equations.
Solution to a SystemAn ordered pair $(x, y)$ that makes ALL equations in the system true. Graphically, it's the point where the lines intersect.For the system $y=x$ and $y=2x-1$, the point $(1,1)$ is a solution because $1=1$ and $1=2(1)-1$ are both true.
Consistent SystemA system of equations that has at least one solution. Graphically, the lines intersect or are the same line.The system $y=x$ and $y=-x+2$ is consistent because they intersect at $(1,1)$, which is a solution.
Inconsistent SystemA system of equations that has no solution. Graphi...
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Core Formulas
Classifying Intersecting Lines (One Solution)
If the graphs of two linear equations intersect at exactly one point, the system has one unique solution.
These lines have different slopes. The system is classified as **consistent** (because it has a solution) and **independent** (because the lines are distinct).
Classifying Parallel Lines (No Solution)
If the graphs of two linear equations are parallel and never intersect, the system has no solution.
These lines have the same slope but different y-intercepts. The system is classified as **inconsistent** (because it has no solution).
Classifying Coincident Lines (Infinitely Many Solutions)
If the graphs of two linear equations are the exact same line (one lies directly on top of the other), the system has infinitely many...
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Challenging
A system is defined by the equations y = ax + 3 and y = 2x + b. For the system to be inconsistent, what must be the values of 'a' and 'b'?
A.a = 2 and b = 3
B.a = 2 and b ≠ 3
C.a ≠ 2 and b = 3
D.a ≠ 2 and b ≠ 3
Challenging
A student incorrectly graphs the line y = 2x + 1 by starting at (0,1) and using a slope of 'up 1, right 2' (rise/run = 1/2). They correctly graph a second line, y = -x + 4. What is the most likely error this student will make when classifying the system?
A.They will correctly classify it as consistent and independent but find the wrong intersection point.
B.They will incorrectly classify it as inconsistent because their lines look parallel.
C.They will incorrectly classify it as consistent and dependent.
D.They will correctly find the intersection point at (1,3) and classify it as consistent and independent.
Challenging
The graph of a system is two coincident lines. If one equation is 3x + y = 5, which of the following could be the other equation?
A.y = -3x - 5
B.y = 3x + 5
C.6x + 2y = 10
D.3x - y = 5
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