Notes
What a solution means
The solution of a system is the point that makes every equation true, where the graphs intersect.
Three ways to solve
- Graphing: read the intersection; best for a quick picture.
- Substitution: best when a variable is already alone, like $y = 3x - 2$.
- Elimination: add or subtract equations (after multiplying, if needed) so one variable cancels.
How many solutions
| Lines | Slopes | Solutions |
|---|---|---|
| Cross once | Different | One |
| Parallel | Same, different intercepts | None |
| Same line | Same, same intercept | Infinitely many |
Inequalities
Graph each boundary (solid for $\le$ or $\ge$, dashed for $<$ or $>$) and shade. Solutions are in the overlap.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Add: $3x = 12$.
$3x + 2(-x + 1) = 6 \Rightarrow x + 2 = 6$.
Same slope, different intercepts.
$x + y = 30$, $x - y = 8$, so $2x = 38$.
$3 \le 4$ and $5 > 3$.
$3(40 - L) + 5L = 156 \Rightarrow 120 + 2L = 156$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A school is choosing between two buses for a field trip. Bus A charges $200 plus $4 per mile. Bus B charges $100 plus $6 per mile.
(a) Write a system of equations for the cost $C$ of each bus for $m$ miles.
Show the answer and scoring guide
Bus A: $C = 200 + 4m$. Bus B: $C = 100 + 6m$.
Check off the points you earned:
(b) Solve the system and explain what the solution means.
Show the answer and scoring guide
$200 + 4m = 100 + 6m \Rightarrow 100 = 2m \Rightarrow m = 50$, $C = 400$. At 50 miles both buses cost $400.
Check off the points you earned:
(c) The trip is 80 miles. Which bus should the school choose? Justify.
Show the answer and scoring guide
Bus A: $200 + 320 = 520$. Bus B: $100 + 480 = 580$. Bus A is cheaper by $60.
Check off the points you earned: