☰ Contents · Algebra

Different methods of solving systems

Lessons 15 · 1 lessons · Sh.A. Alimov, A.R. Xalmukhamedov, M.A. Mirzaakhmedov. Algebra: textbook for Grade 9 of general secondary schools. Revised 4th edition. Tashkent: “O‘qituvchi”, 2019
15

Different methods of solving systems of equations

Textbook: pp. 72–76
GoalApply different methods of solving systems: adding, replacing an expression, symmetric systems and dividing.
New words
term-by-term addition · hadma-had qo‘shishsymmetric system · simmetrik sistemaintroducing a new unknown · yangi noma’lum kiritishdividing one equation by another · tenglamani tenglamaga bo‘lish
Explanation

There are several ways to solve a system, chosen by the form of the equations. 1) Adding or subtracting the equations term by term: this may give expressions such as x + y and xy separately. 2) Replacing an expression: if y² or xy can be expressed through x from one equation, substitute it into the other. 3) Symmetric systems (unchanged when x and y are swapped): put s = x + y, p = xy; then x² + y² = s² − 2p, because (x + y)² = x² + 2xy + y². Then solve z² − sz + p = 0. 4) Dividing one equation by the other (after checking x ≠ 0, y ≠ 0), for example from x²y = 4 and xy² = 2 we get x : y = 2. Always check the pairs found, especially with roots or fractions.

Worked examples
x² + xy = 15, y² + xy = 10: adding gives (x + y)² = 25, x + y = ±5; subtracting gives x² − y² = 5, i.e. (x − y)(x + y) = 5, so x − y = ±1. From x + y = 5, x − y = 1 we get (3, 2); from x + y = −5, x − y = −1 we get (−3, −2).
x + y² = 5, xy² = 4: y² = 5 − x, x(5 − x) = 4, x² − 5x + 4 = 0, x₁ = 1, x₂ = 4. For x = 1, y² = 4, y = ±2; for x = 4, y² = 1, y = ±1. Answer: (1, 2), (1, −2), (4, 1), (4, −1).
Class activity

“Choose the method”: the teacher hands out cards with systems; groups first say which method is convenient (adding, replacing, s and p, dividing), then solve and check the answers.

Practice
1
Solve the system x² + y² = 34, xy = 15.
2
Solve the system x²y = 4, xy² = 2.
3
Solve the system 1/x + 1/y = 3/4, xy = 8.
4
What must be checked before dividing one equation by another, and why?