Key Terms
Independent system
Exactly one solution; lines intersect at one point.
Example
70 mL of 50% solution + x mL of 80% solution = final 60% solution 35 + 0.8x = (70 + x)(0.6) 35 + 0.8x = 42 + 0.6x 0.2x =
Steps
1. Line up like terms vertically.
Example - no multiplication needed
X + 2y = -1 -x + y = 3 Add: 3y = 2, so y = 2/3 Back-substitute to get x = -7/3 Solution: (-7/3, 2/3)
Example - one equation needs multiplication
3x + 5y = -11 and x - 2y = 11 Multiply second equation by -3: -3x + 6y = -33 Add: 11y = -44, y = -4
Back-substitute
X = 3 Solution: (3, -4)
Example - both equations need multiplication
2x + 3y = -16 and 5x - 10y = 30 Multiply first by -5: -10x - 15y = 80 Multiply second by 2: 10x - 20y = 60 Add: -35y = 1
Quick Reference
What Your Algebra Tells You
Revenue function
R = xp (quantity times price).
Break-even point
Where cost equals revenue; profit equals zero.
Left of break-even
Operating at a loss. Right of break-even: operating at a profit.
Use a three-row table
Start | Add | Final. Columns: Amount | Part (concentration or price) | Total (Amount times Part).
System of linear equations
Two or more linear equations in two or more variables solved simultaneously.
Dependent system
Infinite solutions; equations represent the same line.
Inconsistent system
No solution; equations represent parallel lines.