Unit 4 · Core unit
Two conditions holding at once. The number of solutions is a geometric fact about whether the lines meet, are parallel, or coincide.
What a strong answer looks like
A strong answer says how many solutions exist and why, before or alongside finding them.
Know
Substitution replaces one variable using the other equation; elimination scales equations so adding removes a variable. Both are valid and one is usually far less work.
Apply
Choose substitution when a variable is already isolated, elimination when coefficients align easily.
Watch out
Substituting into the same equation it came from, which yields a true statement and no information.
Study move
Solve one system both ways and compare the effort.
Know
Equal slopes with different intercepts give no solution; identical equations give infinitely many; otherwise there is exactly one.
Apply
Compare slopes and intercepts before solving to predict the outcome.
Watch out
Reporting no solution when the algebra produced an always-true statement, which means infinitely many.
Study move
Construct one system of each of the three kinds.
Know
The solution is the region where every individual solution region overlaps.
Apply
Graph each inequality, shade, and identify the common region including whether boundaries are included.
Watch out
Forgetting that a strict inequality excludes its boundary line.
Study move
Graph a two-inequality system and describe the region in words.
Modeling drill
Two ticket types total a known count and a known revenue. Set up and solve the system, then say what each answer means.
