Notes
Solving systems
- A solution makes all equations true. Graphing: find the intersection. Substitution: replace a variable with an equivalent expression. Elimination: add or subtract equations to cancel a variable.
- Number of solutions: one (crossing lines), none (parallel lines, false statement like $0 = 5$), or infinitely many (same line, true statement like $0 = 0$).
Modeling
- Define two variables, write two equations, solve, and interpret. Common cases: two cost plans, ticket counts and prices, sum and difference.
Systems of inequalities
- Graph each boundary (dashed for $<$, $>$; solid for $\leq$, $\geq$), shade the true side, and use the overlap. Test points to check.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$2x + x + 4 = 10$ gives $x = 2$, so $y = 6$.
Add: $6x = 12$ gives $x = 2$. Then $y = 3 - 4 = -1$.
The slopes are equal and the intercepts differ, so the lines are parallel.
$s + a = 12$ and $6s + 10a = 88$. Then $72 + 4a = 88$, so $a = 4$ and $s = 8$.
At $(1, 1)$: $1 < 3$ and $1 \geq -1$. Each other point fails one inequality.
The $y$-coefficients $3$ and $-3$ are opposites, so they cancel when added.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Bike shop X charges a $60 enrollment fee plus $20 per month. Bike shop Y charges no enrollment fee and $35 per month. Let $m$ be the number of months and $C$ the total cost.
(a) Write a system of equations for the two shops.
Show the answer and scoring guide
Shop X: $C = 60 + 20m$. Shop Y: $C = 35m$.
Check off the points you earned:
(b) After how many months do both shops cost the same, and what is that cost?
Show the answer and scoring guide
$60 + 20m = 35m \Rightarrow 60 = 15m \Rightarrow m = 4$. The cost is $35(4) = 140$, so $140.
Check off the points you earned:
(c) A student plans 10 months. Which shop is cheaper and by how much?
Show the answer and scoring guide
Shop X: $60 + 20(10) = 260$. Shop Y: $35(10) = 350$. Shop X is cheaper by $90.
Check off the points you earned: