## 1. NOTION OF A LINEAR EQUATION

If A* *and

*B are algebraic expressions if at least one of them contains a variable, then the formula (equation)*

**B is call**

*A =**ed an algebraic equ*ation; the variables are said to be

*unknown i*n the equation. An algebraic equation can:

– to

**on, ie. i**

*have a soluti**s a solvable e*quation, if its set of solutions is not an empty set;

– to

**on, ie. i**

*have no soluti**s an unsolvable equ*ation if there is no solution.

Two equations* A = *B

*= D are said t*

**and C***uations if their sets of solutions match. Denotes: (A*

**o be equivalent eq**

**= B <=> C = D) <=> M₁ = M₂**where M1 i

**the set of solutions of the equation A = B**

*s***nd MZ**

*, a**the equation C*

**o**f*D.*

**=**## 2. LINEAR EQUATIONS WITH ONE UNKNOWN

If in the algebraic equation* A = *B after its arrangement the unknown x appears only with the first degree, then that equation is said to be

*a linear equatio*n. Each linear equation can be brought to the form

**a**0, where a

*h + b =***d**

*a*n**are some real numbers. The equati**

*b**= 0, in case when:*

**on ax + b**1 °. ** a ≠ 0**, it is solvable and has only

*one solution:*

**– b / a**2 °.* a = 0 ∧ *b = 0, It is also solvable, ie. every real number is its solution, ie ther

*e are infinitely many solutions*

3 °. ** a = 0 ∧ b ≠ **0, the equation is unsolvable, because the equatio

*0, does not become an exact numerical equation for any real number*

**n 0 • x + b**= 0**, b ≠***x.*

## 3. LINEAR EQUATION WITH TWO UNKNOWN

Any algebraic equation that can be brought into the form:

* ax + by = c* (1)

where a, * b, c ar*e real numbers is called a

*linear equation with two variable*s (unknown) x

*nd*

**a****. The number**

*y**a a*

**s****d b a**

*n**re coefficients before the unkno*wns,

*nd c is a free term of the equation.*

**a**The solution of equation (1) is any ordered pair of real numbers

*x*

**(***уₒ) for which it becomes an exact equation.*

**ₒ,**The equat

*y = c, in case a*

**ion ax + b***or b*

**а 0****has infinitely many solutions**

*≠ 0*In case a = 0 an

*=*

**d b***he equation a*

**0, t***of the form 0*

**x + by = c i**s*nd for:*

**x + 0y = c a**it is unsolvable a**c ≠ 0**d M**n***is an empty set*,it is solvable and ha**c = 0***s infinitely many solution*s*(M = R x*R).

The graphical representation of a linear equation with two variables is straight.

## 4. LINEAR INEQUALITY WITH AN UNKNOWN

If two algebraic expressio* n*s A

*and B (which have one variable each) are associated with one of the signs for*

*ity*

**inequa**l*, ≤,> or ≥*

**<***ple: A <B), then we get a predicate with one variable called*

**(exam***inequality with one unknown.*

The inequali* ties *A <

*C <D are equivalent if the set of solutions is equal to the set of solutions of the other inequality, ie*

**B and****A <B <=> C <D**

Properties of inequalities

–** A <B <=> B> A**

–** A <B <=> A + C <B + C – C> 0 => (A <B <=> AC <BC)**

–** C <0 => (A <B <=> AC> BC)**

Any inequality that can be converted to the form ah <b, where a and b are real numbers, is called* a linear inequality with an unknown.*

## 5. GRAPHIC SOLUTION OF LINEAR INEQUALITIES WITH TWO VARIABLES

The shape inequality:

* ax + by + c> 0*, or a

*ie ax + b*

**x + by + c <0,***ax + b*

**y + c ≥ 0, or***re a, b, c*

**y + c ≤ 0, whe***и у*

**∈ R, а х***се*

**ariables, is called a linear inequality with two variables. The solutions of a linear inequality with two variables is the set of points of one half plane.**

*v*** Example 1.** The set of solutions of the inequalit**y 2x – u + 4> 0**

is graphically specified in Figure 1 with a half-hatched section.

Fig.1