	Algebra Tutorials!



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	Solving Quadratic Equations by Completing the Square
	Graphing Logarithmic Functions
	Division Property of Exponents
	Adding and Subtracting Rational Expressions With Like Denominators
	Rationalizing the Denominator
	Multiplying Special Polynomials
	Functions
	Solving Linear Systems of Equations by Elimination
	Solving Systems of Equation by Substitution and Elimination
	Polynomial Equations
	Solving Linear Systems of Equations by Graphing
	Quadratic Functions
	Solving Proportions
	Parallel and Perpendicular Lines
	Simplifying Square Roots
	Simplifying Fractions
	Adding and Subtracting Fractions
	Adding and Subtracting Fractions
	Solving Linear Equations
	Inequalities in one Variable
	Recognizing Polynomial Equations from their Graphs
	Scientific Notation
	Factoring a Sum or Difference of Two Cubes
	Solving Nonlinear Equations by Substitution
	Solving Systems of Linear Inequalities
	Arithmetics with Decimals
	Finding the Equation of an Inverse Function
	Plotting Points in the Coordinate Plane
	The Product of the Roots of a Quadratic
	Powers
	Solving Quadratic Equations by Completing the Square

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Solving Systems of Equation by Substitution and Elimination

The problems with the graphing method are threefold:

You need an accurate graph.

Your graph may not be large enough.

It’s hard to estimate the solution if the coordinates are not integers.

In this section, we look at two algebraic methods for finding solutions.

In both of the methods outline below, there are actually three possible outcomes.

You get a single ordered pair as a solution.

In this case, the solution is the ordered pair you find.

All variables go away and you get a false statement, such as 0 = 4.

In this case, you have parallel lines, so there is no solution to the system.

All variables go away and you get a true statement, such as 0 = 0 or 5 = 5.

In this case, you have the same line, so there are infinitely many solutions.

Substitution

Procedure: (Substitution Method)

0. Choose a variable and an equation.

1. Solve for the chosen variable in the chosen equation.

2. Substitute the expression you found for the selected variable in the OTHER equation.

3. Solve the resulting equation in one variable.

4. Use the answer you found in 3 to find the value of the other variable.

5. Write your answer as an ordered pair.

Example:

x + y = 8

2x - 3y = -9

Answer:

(3, 5).

Elimination

Procedure: (Elimination Method)

0. Choose a variable.

1. Multiply one or both equations by whatever is necessary to get the coefficients of the selected variable to be the same, but with opposite signs.

2. Add the equations together. (NOTE: This eliminates the selected variable.)