Algebra Mastery

Basic Concepts of Algebra

• Elementary algebra introduces variables (quantities without fixed values).

• It builds on arithmetic and is taught to secondary school students.

• Algebraic notation includes coefficients, terms, variables, and constants.

• Algebraic operations include addition, subtraction, multiplication, division, and exponentiation.

• Alternative notation is used when formatting is not available.

• Variables represent general (non-specified) numbers.

• Algebraic expressions may be evaluated and simplified.

• Equations use the symbol for equality (=) and state that two expressions are equal.

• Conditional equations are true for only some values of the involved variables.

• An identity equation is true for all values of the involved variables.

• Algebra is used to express many quantitative relationships in science and mathematics.

• Elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.Understanding Algebraic Equations

• Algebraic equations are mathematical expressions that state the equality of two or more unknown variables.

• Equations can be solved by finding the values of the variables that make the equation true.

• Inequalities are used to show that one side of the equation is greater or less than the other.

• The properties of equality include reflexivity, symmetry, and transitivity.

• Substitution is replacing the terms in an expression to create a new expression or statement.

• Solving algebraic equations requires isolating the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

• Linear equations with one variable contain only constant numbers and a single variable without an exponent.

• The general form of a linear equation with one variable can be written as ax+b=c.

• Linear equations with two variables require two related equations to solve.

• Quadratic equations include a term with an exponent of 2 and can be solved using the quadratic formula.

• The quadratic formula is x = (-b ± sqrt(b^2-4ac))/2a.Algebraic Equations Summary

• A quadratic equation is a second-degree polynomial equation in a single variable.

• A quadratic equation can be expressed in the form ax² + bx + c = 0, where a, b, and c are coefficients.

• All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system.

• Quadratic equations can be solved using the quadratic formula or factorization.

• An exponential equation is one that has the form a^x = b, and it has a solution of x = log_a(b).

• A logarithmic equation is one that has the form log_a(x) = b, and it has a solution of x = a^b.

• A radical equation is one that includes a radical sign, which includes square roots, cube roots, and nth roots.

• An nth root can be rewritten in exponential format, so that x^(1/n) is equivalent to the nth root of x.

• A quadratic equation must contain the term ax², which is known as the quadratic term.

• Quadratic equations can be solved using completing the square, which leads to the quadratic formula.

• A quadratic equation can have a root of multiplicity 2, which means the root appears twice.

• Complex numbers first arise in the teaching of quadratic equations and the quadratic formula.Solving Linear Equations with Two Variables

• A radical equation is of the form x^(3/2) or (x^m/n). It can be solved using the formula sqrt[n]{x^m}=a.

• A system of linear equations can be solved using different methods, including elimination and substitution.

• In the elimination method, equations are added or subtracted to eliminate one variable and solve for the other.

• In the substitution method, one variable is isolated from one equation and substituted into the other equation.

• Some systems of linear equations are inconsistent, meaning they have no solution. An example is 0x+2y=3 and 0x+2y=4.

• Some systems of linear equations are undetermined, meaning they have infinitely many solutions. An example is x+2y=6 and 2x+4y=12.

• Some systems of linear equations are underdetermined, meaning they have more variables than equations. An example is x+2y+z=6 and x+y=3.

• Some systems of linear equations are overdetermined, meaning they have more equations than variables. If they have any solutions, some equations are linear combinations of the others.