Linear equations and inequalities

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) A = B is called an algebraic equation; the variables are said to be unknown in the equation. An algebraic equation can:
– to have a solution, ie. is a solvable equation, if its set of solutions is not an empty set;
– to have no solution, ie. is an unsolvable equation if there is no solution.

Two equations A = B and C = D are said to be equivalent equations if their sets of solutions match. Denotes: (A = B <=> C = D) <=> M₁ = M₂
where M1 is the set of solutions of the equation A = B, and MZ of the equation C = D.

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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 equation. Each linear equation can be brought to the form ah + b = 0, where a and b are some real numbers. The equation ax + b = 0, in case when:

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 there are infinitely many solutions

3 °. a = 0 ∧ b ≠ 0, the equation is unsolvable, because the equation 0 • x + b = 0, b ≠ 0, does not become an exact numerical equation for any real number x.

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3. LINEAR EQUATION WITH TWO UNKNOWN

Any algebraic equation that can be brought into the form:

ax + by = c (1)

where a, b, c are real numbers is called a linear equation with two variables (unknown) x and y. The numbers a and b are coefficients before the unknowns, and c is a free term of the equation.
The solution of equation (1) is any ordered pair of real numbers (xₒ, уₒ) for which it becomes an exact equation.
The equation ax + by = c, in case a а 0 or b ≠ 0 has infinitely many solutions
In case a = 0 and b = 0, the equation ax + by = c is of the form 0x + 0y = c and for:

  • c ≠ 0 it is unsolvable and M is an empty set,
  • c = 0 it is solvable and has infinitely many solutions (M = R x R).

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

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4. LINEAR INEQUALITY WITH AN UNKNOWN

If two algebraic expressions A and B (which have one variable each) are associated with one of the signs for inequality <, ≤,> or ≥ (example: A <B), then we get a predicate with one variable called inequality with one unknown.

The inequalities A <B and C <D are equivalent if the set of solutions is equal to the set of solutions of the other inequality, ie

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.

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5. GRAPHIC SOLUTION OF LINEAR INEQUALITIES WITH TWO VARIABLES

The shape inequality:

ax + by + c> 0, or ax + by + c <0, ie ax + by + c ≥ 0, or ax + by + c ≤ 0, where a, b, c ∈ R, а х и у се variables, 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.

Example 1. The set of solutions of the inequality 2x – u + 4> 0

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

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Fig.1

– Linear equations and non-equations –
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