Mathematics
Grade 5
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
Identify and plot points on a coordinate plane from a given rule or table.
Graph two simple relationships on the same coordinate plane.
Determine if two graphed lines intersect, are parallel, or are the same line.
Find the point of intersection of two lines on a graph and explain what it means in a real-world context.
Explain what it means when two lines are parallel on a graph in a real-world context.
Classify a system of two relationships as having one solution, no solution, or many solutions based on their graphs.
Have you ever wondered when two different plans or deals might end up costing the same amount? 💰 Let's use graphs to find out!
In this lesson, you'll learn how to compare two different rules or patterns by drawing them on a graph....
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Key Concepts & Vocabulary
TermDefinitionExample
Coordinate PlaneA flat surface made of two number lines (x-axis and y-axis) that cross each other, used to plot points.Plotting the point (3, 2) means going 3 units right on the x-axis and 2 units up on the y-axis.
Ordered PairTwo numbers written in a specific order, like (x, y), that tell you the exact location of a point on a coordinate plane.In the ordered pair (5, 10), 5 is the x-coordinate and 10 is the y-coordinate.
Graphing a RelationshipDrawing a picture on a coordinate plane that shows how two quantities are connected by a rule or pattern.If the rule is 'cost = number of items x $2', you can graph points like (1, $2), (2, $4), (3, $6).
System of RelationshipsWhen we look at two different rules or patterns together to compare them and see how they r...
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Core Formulas
Classifying by Intersection (One Solution)
If two graphed lines cross at exactly one point, the system of relationships has one solution. This point is the common solution.
Use this rule when you see two lines meet. The coordinates of the meeting point tell you the specific values where both rules are true.
Classifying by Parallel Lines (No Solution)
If two graphed lines are parallel and never cross, the system of relationships has no solution.
Use this rule when you see two lines that stay the same distance apart. This means there are no values where both rules give the same result.
Classifying by Same Line (Many Solutions)
If two graphed lines lie exactly on top of each other, the system of relationships has infinitely many solutions.
Use this rule when two differe...
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Challenging
A graph shows one line for the rule y = x + 3. Which rule for a second line would create a system with NO solution?
A.y = 2x + 3
B.y = x + 5
C.y = 3x
D.y = x + 3
Challenging
A student correctly plots the line for the cost of apples at a farm: Cost = $2 per pound. The line goes through (0,0), (1,2), and (2,4). A second line on the same graph represents the cost of pears. This second line goes through (0,1), (1,3), and (2,5). Without finishing the graph, what can you conclude?
A.The lines will intersect because the cost per pound is different.
B.The lines are the same, so there are many solutions.
C.The lines are parallel, so there is no solution.
D.The lines will intersect at the origin.
Challenging
A student claims: 'If two different pizza places have different delivery fees but charge the same price per pizza, their total cost graphs will eventually intersect.' Is this claim correct?
A.Yes, because the total cost will eventually be high enough to meet.
B.Yes, but only if you order a very large number of pizzas.
C.No, because the lines will be parallel and never intersect.
D.No, because one line will always be below the other but they will get closer.
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