Maryland Math 8.AT.B.5.c
The Standard
- 8.AT.B.5
- Analyze and solve systems of two linear equations in two variables.
Use structure to determine when a system has no solution, one solution, or infinitely many solutions and justify the conclusion.
Maryland College and Career Ready Standards for Mathematics (2025) · MAKE SENSE OF AND SOLVE EQUATIONS, INEQUALITIES, AND SYSTEMS OF EQUATIONS.
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students decide how many solutions a pair of linear equations has. They compare slopes and starting values, or check whether one whole equation is a multiple of the other. They explain why the lines cross, stay apart, or match.
What Mastery Looks Like
- A student spots that 4x+2y=12 is twice 2x+y=6, so the lines match. They identify equal slopes with different starting values as no solution. They identify different slopes as one solution and explain why.
Common Misconceptions
- Some students think two equations must have one solution because there are two unknowns. Others multiply the x and y terms but forget the constant, so they call parallel lines the same line. Some mistake a shared starting value for infinitely many solutions, even when the slopes differ.
How to Assess It
- Exit ticket: Classify each pair as no solution, one solution, or infinitely many solutions, and justify without graphing: (1) y=2x+3 and y=2x-1; (2) 2x+y=6 and 4x+2y=12; (3) y=2x+3 and y=-x+3.
Lesson moves
Ways to Teach It
Draw y=x+1 twice, y=x-1, and y=-x+1 on clear grid sheets; overlay pairs and explain each intersection count.
Ask, "Why do 2x+y=6 and 4x+2y=12 have infinitely many solutions even though they look different?" Require an explanation using the equations.
Sort cards showing equation pairs into no, one, or infinitely many solutions; write a slope or multiplication clue on each card.
Compare bike rental plans charging $5 plus $3 per hour and $9 plus $2 per hour; find when the prices match.
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