Basic algebra

• To check if a solution is correct, substitute the value of the variable back into the original equation and see if both sides are equal.
• Subtracting Real Numbers involves changing the subtraction symbol to the addition symbol, changing the sign of the number being subtracted, and adding using the rules for adding real numbers.
• Multiplying Real Numbers involves multiplying the absolute value of the two numbers, and if the two numbers have the same sign, the product is positive, and if the two numbers have different signs, the product is negative.
• Division by 0 is undefined.
• Dividing Real Numbers involves dividing the absolute value of the numbers, and if the signs are the same, the answer is positive, and if the signs are different, the answer is negative.
• Commutative Properties, Associative Properties, Distributive Properties, Identity Properties, Inverse Properties, and Simplifying Algebraic Expressions are properties of real numbers.
• When adding or subtracting algebraic expressions, only like terms can be combined.
• To add or subtract fractions with the same denominator, add or subtract the numerators and keep the same denominator, and to add or subtract fractions with different denominators, find the LCD and write each fraction with this LCD.
• Multiplying numerators and multiplying denominators is a part of Multiplying Real Numbers.
• Multiplying the first fraction by the reciprocal of the second fraction is a part of Division.
• Simplify within parentheses, brackets, or absolute value bars or above and below fraction bars first, in the following order: apply all exponents, perform any multiplications or divisions from left to right, perform any additions or subtractions from left to right.
• To find the value(s) for which a rational expression is undefined, set the denominator equal to 0 and solve the resulting equation.
• Three variables can be handled using the elimination method by eliminating any variable from any two of the original equations, eliminating the same variable from any other two equations, and using the elimination method for two-variable systems to solve for the two variables.
• Dividing rational expressions involves multiplying the first rational expression by the reciprocal of the second rational expression, factoring numerators and denominators, and writing the expression in lowest terms.
• Multiplying rational expressions involves multiplying numerators and multiplying denominators, factoring numerators and denominators, and writing the expression in lowest terms.
• The elimination method for systems of linear equations involves writing the equations in standard form, multiplying one or both equations by appropriate numbers so that the sum of the coefficient of one variable is 0, adding the equations to eliminate one of the variables, and solving the equation that results from the addition.
• To write a rational expression in lowest terms, factor the numerator and denominator, divide out common factors, and write the expression in lowest terms.
• Writing a rational expression with a specified denominator involves factoring both denominators, determining what factors the given denominator must be multiplied by to equal the one given, multiplying the rational expression by that factor divided by itself, and writing the expression in lowest terms.
• Finding the least common denominator (LCD) involves factoring each denominator into prime factors, listing each different factor the greatest number of times it appears in any one denominator, multiplying the factors, and writing the expression in lowest terms.
• Adding or subtracting rational expressions involves finding the LCD, rewriting each rational expression with the LCD as denominator, and solving the equation that results from the LCD.
• An expression containing a variable is evaluated by substituting a given number for the variable, and values for a variable that make an equation true are solutions of the equation.
• a is less than b if a is to the left of b on the number line.
• The additive inverse of x is -x.
• The absolute value of x, denoted | x |, is the distance (a positive number) between x and 0 on the number line.
• Multiplying Complex Numbers: Multiply using FOIL expansion and using to reduce the result.
• The solutions of , given by the discriminant , determine the number and type of solutions.
• To add two numbers with the same sign, add their absolute values, and the sum has the same sign as each of the numbers being added.
• The graph of a quadratic function is a parabola, opening up if positive, down if negative.
• Solving Quadratic Equations by Completing the Square: To solve , divide each side by and write the equation with the variable terms on one side of the equals sign and the constant on the other.
• Solving Quadratic Equations: Square Root Property: If a is a complex number, then the solutions to are and .
• To multiply radicals with negative radicands, first change each factor to the form A complex number has the form , where a and b are real numbers.
• The vertex of a parabola is the point where the axis of symmetry intersects the curve.
• The axis of symmetry of a parabola is the line that divides the curve into two symmetrical halves.
• The equation that defines the inverse function f –1 is found by interchanging x and y, solving for y, and replacing y with f –1 ( x ).
• If any horizontal line intersects the graph of a function in, at most, one point, then the function is one to one and has an inverse.
• The graph of a horizontal parabola opens to the right if positive, to the left if negative.
• Dividing Complex Numbers: Multiply the numerator and the denominator by the conjugate of the denominator.
• Adding and Subtracting Complex Numbers: Add (or subtract) the real parts and add (or subtract) the imaginary parts.
• Use the square root property to determine the solutions.
• Factor the perfect square trinomial and write it as the square of a binomial.