30. Two unknowns: three ways to eliminate

The lesson, step by step

1. Warm-up

   Ask: If x + y = 10 and x − y = 2, what are x and y? (6 and 4; let your student guess and check.)

2. Elimination

   Read Art. 157.

   Elimination is the process of deducing from two or more equations containing two or more unknown quantities, a less number of equations containing one less unknown quantity.

3. Substitution

   Read Art. 158 and its rule:

   1. Find an expression for the value of one of the unknown quantities in either equation.

   2. Substitute this value in place of the same unknown quantity in the other equation; there will thus be formed a new equation containing only one unknown quantity.

   Work the book's pair: x = 17 − 2y, so 2(17 − 2y) + 3y = 28, y = 6, x = 5.

4. Comparison

   Read Art. 159.

   Elimination by Comparison consists in finding the value of the same unknown quantity in two different equations, and then placing these values equal to each other.

   Solve the same pair this way and get the same answer.

5. Addition and subtraction

   Read Art. 160 and its rule:

   1. Multiply or divide the equations, if necessary, so that one of the unknown quantities will have the same coefficient in both.

   2. Add or subtract the equations, according as the signs of the equal terms are alike or unlike, and the resulting equation will contain only one unknown quantity.

   Work 3x − 5y = 6 and 4x + 3y = 37 together (x = 7, y = 3).

6. Guided and independent practice

   Your student does exercise 3 of Art. 158 (3x + 5y = 57, 5x + 3y = 47; x = 4, y = 9) and exercise 1 of Art. 160 (3x + 2y = 21, x − 2y = −1; x = 5, y = 3) while you watch, then the rest of the exercises under all three articles.

7. Check

   Substitute both values into BOTH equations. Afterwards, one can be confirmed on the site's calculator (/homeschool/calculator/) with solve.

8. Wrap-up

   Ask: Which method did you like best? The book says addition and subtraction is generally preferred.