Ray's New Elementary Algebra · Art. 219 · Unit 7: Chapter VII: Quadratic equations
50. Quadratic equations with two unknowns
Goal: Your student solves pairs like x + y = 8 and xy = 15 by substitution and by the sum-and-difference method.
You'll need
- The book, open to the page
- Pencil and plenty of paper (or a whiteboard)
- A card to cover printed answers
What's in the book
Simple methods for pairs of equations of the second degree, worked examples 1 to 5 and exercises 6 to 19; then fourteen problems with two unknowns. Answers are printed. A general review follows.
The lesson, step by step
Warm-up
Ask: What two numbers add to 8 and multiply to 15? (5 and 3.)
Substitution
Read example 2: from x + y = 8, x = 8 − y; then y(8 − y) = 15, a quadratic giving y = 5 or 3.
The general method
Read the method on pp. 210 and 211: square x + y = a, subtract 4xy = 4b, and take the root to get x − y. Then add and subtract.
Guided practice
Your student solves while you watch:
7. x+y=16, xy=63 Ans. x=9, or 7.
Then exercise 9: x + y = 9 and x² + y² = 53 (7 and 2).
Problems
Your student sets up and solves:
1. The sum of two numbers is 10, and the sum of their squares 52 ; what are the numbers? Ans. 4 and 6.
2. The difference of two numbers is 3, and the difference of their squares 39; required the numbers. Ans. 8 and 5.
Independent practice
Exercises 6, 8, 10 to 12, and problems 5 to 8.
Wrap-up
Use the General Review on p. 215 as an oral quiz on the whole book so far.
Tip
The book itself says a full treatment belongs in a higher algebra. Do the simpler ones well rather than all of them.
Answers to the book's problems
The book prints the answer after each exercise.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.