Ray's New Elementary Algebra · Art. 155 · Unit 4: Chapter IV: Simple equations
26. Solving simple equations
Goal: Your student solves first-degree equations in one unknown by the four-step rule, and verifies every answer.
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
How to undo addition, subtraction, multiplication and division, worked examples 5 to 7 (including verification), the rule, and 32 examples for practice. Answers are printed.
The lesson, step by step
Warm-up
Solve x + 3 = 5, x − 3 = 5, 3x = 15 and x/3 = 2 (the book's four opening examples).
The pattern
Read the four statements on p. 96.
When the unknown quantity is connected by addition, it is to be separated by subtraction.
Your student says the other three in their own words.
Work an example together
Example 5: 3x − 3 = x + 5. Transpose: 3x − x = 5 + 3. Reduce: 2x = 8. Divide: x = 4. Verify: 3 × 4 − 3 = 4 + 5, or 9 = 9.
This method of substituting the value of the unknown quantity instead of itself, is called verification.
The rule
1. If necessary, clear the equation of fractions.
2. Transpose all the terms containing the unknown quantity to one side, and the known quantities to the other.
3. Combine the terms in each member by the rule for addition.
4. Divide both members by the coefficient of the unknown quantity.
Guided practice
2. 3x— 8=16— 5x Ans. x=3.
5. 5(x+1)+6(x+2)=9(x+3) Ans. x=5.
Your student works both, saying the step names aloud, and verifies.
Independent practice
Examples 1 to 19 today (the numerical ones). Save the literal ones (20 to 32) for another session or as a challenge.
Check
Verification is the check: substitute every answer back. After that, your student may also confirm one on the site's calculator (/homeschool/calculator/) with solve(5*(x+1)+6*(x+2)=9*(x+3), x).
Wrap-up
Ask: Why do we verify, if we already followed the rule carefully?
Afterwards
Examples 20 to 32 (letters for known quantities) make a good second session. Example 21, ax + b = cx + d, gives x = (d − b)/(a − c).
Tip
Insist on verification every time, as Ray does. It builds independence: your student knows the answer is right without asking.
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 ↗.