Algebraic Equations and Systems: Core Concepts
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Identities and Equations
An algebraic equality is formed by two algebraic expressions separated by an equal sign. They are of two types:
- Identity: True for any value of the letters.
- Equation: True only for specific values of the letters.
Elements of an Equation
- Member: Each of the two algebraic expressions separated by an equal sign.
- Term: Each summand in each of the members.
- Unknowns: Letters whose values are unknown.
- Degree: The greatest exponent of the unknowns.
The values of the unknowns that make the equation true are called solutions. Two equations are equivalent when they have the same solution.
2nd Degree Equations
A 2nd degree equation with one unknown is an equality of the form ax2 + bx + c = 0, where a ≠ 0.
Analyzing the Number of Solutions
The discriminant is defined as: Δ = b2 - 4ac. The number of solutions depends on the value of the discriminant:
- If Δ > 0, the equation has two distinct real solutions.
- If Δ = 0, the equation has a double real solution.
- If Δ < 0, the equation has no real solutions.
Equations with Two Unknowns
Linear equations with two unknowns are of the form ax + by = c. The solutions are pairs of numbers that satisfy the equation.
Systems of Equations
Two linear equations with two unknowns form a system where we seek common solutions:
- Incompatible System: No solution; graphically represented as two parallel lines.
- Compatible Determined System: Has a unique solution; graphically represented as two intersecting lines.
- Compatible Undetermined System: Has infinite solutions; graphically represented as two coincident lines.
Methods of Solving Systems
- Substitution Method: Solve for one unknown in one equation and substitute it into the other.
- Equalization Method: Solve for the same unknown in both equations and equate the resulting expressions.
- Reduction Method: Prepare the equations so that one unknown has the same coefficient but opposite signs in both. Adding the equations member-by-member results in a single equation with one unknown.