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Pre-algebra                        Arithmetics                                         Integers                                         Divisibility                                         Decimals                                         Fractions                                         Exponents                                         Percentages                                         Proportional reasoning                                         Radical expressions                                         Graphs                 		                                               Algebra                        Monomials                                         Polynomials                                           Factoring                                   Linear Equations                                           Graphs of linear equations                                Rectangular Coordinate System                 Midpoint Formula                 Definition of Slope                 Positive and negative slope                 Determine the slope of a line                 Equations of lines                                                       Equation of lines (from graph)                 Applications of linear equations                                         Inequalities                                         Quadratic equations                                           Graphs of quadratic equations                                           Absolute Value                                         Radical expressions                                         Exponential equations             Logarithmic equations             System of equations                                         Graphs and functions                                    Plotting points and naming quadrants                   Interpreting Graphs                   Relations and Functions                   Function Notation                   Writing a Linear Equation from a Table                   Writing a Linear Equation to describe a Graph                   Direct Variation                   Indirect Variation                   Domain and range                                                            Sequences and series                                         Matrices                                         Inverse of a matrix             Determinants             Inner product                                  Geometry                            Triangles                                                 Polygons                                               2-D Shapes                                                 3-D Shapes                                               Areas                                                 Volume                                     Pythagorean Theorem                                   			                 Angles               Building Blocks                                                 Geometry Transformations                                               Parallel, coincident and intersepting lines               Distances in the plane                                                 Lines in space                                               Plane in space                                               Angles in the space                                                 Distances in the space                                                 Similarity                          Precalculus                        Sequences and series                                           Graphs                                    Graphs                   Definition of slope                   Positive or negative slope                   Determine the slope of a line                   Equation of a line (slope-intercept form)                   Equation of a line (point slope form)                   Equation of a line from graph                   Domain and range                                      Quadratic function                   Limits (approaches a constant)                   Limits (approaches infinity)                   Asymptotes                   Continuity and discontinuities                                         Parallel, coincident and intersepting lines             Introduction to Functions                                     Limits                                           Continuity                                           Asymptotes                                                     Trigonometry                        Trigonometric ratios                                           The reciprocal trigonometric ratios                                           Trigonometric ratios of related angles                                           Trigonometric identities             Solving right angles                                           Law of sines             Law of cosines             Domain of trigonometric functions                         Statistics                    Mean           Median           Mode           Quartiles           Deciles           Percentiles           Mean deviation           Variance           Standard Deviation           Coefficient of variation           Skewness           kurtosis           Frequency distribution                                     Graphing statistics & Data                                     Factorial           Variations without repetition           Variations with repetition           Permutations without repetition           Permutation with repetition           Circular permutation           Binomial coefficient           Combinations without repetition           Combinations with repetition

Simultaneous equations
Systems of 3 variable Equations

Systems of equations that have three variables are systems of planes. Like systems of linear equations, the solution of a system of planes can be no solution, one solution or infinite solutions.

To solve a system of, write the given system in matrix form. Write all the information contained in the system in the single augmented matrix and choose one of these method to solve the system:

• Gaussian Elimination: Take an augmented matrix for the system and carry it by means of elementary row operations to an equivalent augmented matrix from which the solutions of the system are easily obtained. In particular, we bring the augmented matrix to Row-Echelon Form. At that point, the solutions of the system are easily obtained.

• Matrix inverse method: Write the given system as a single matrix equation AX=B . Solve the equation obtained for the matrix variable X by left-multiplying both sides of the above matrix equation (AX=B) by the inverse of the matrix A. Then, and you have the solution.

• Cramer's Rule: To use this method, you have only to follow the steps:

1. Write the coefficient matrix of the system (call this matrix A); if it is square, you may continue, otherwise Cramer's rule is not applicable here.
2. Compute the determinant of the coefficient matrix, |A|; if |A| is not zero you may continue, otherwise Cramer's rule is not applicable here.
3. Suppose the first variable of the system is x. Then write the matrix Ax as follows: substitute the column of numbers to the right of the equal signs instead of the first (from the left) column of A. Now compute the determinant of Ax, that is |Ax|.
4. The value of x in the solution is now |Ax| / |A|.
5. Repeat steps 3, and 4 with the remaining variables. In each case substitute the column of numbers instead of the column of A that corresponds to the variable you are using. If the variables are x, y, and z, then the values will now be:

x = |Ax| /   |A|  y = |Ay| / |A|  z = |Az| /   |A|

And the solution is

(x, y, z) = ( |Ax| / |A| ,   |Ay| / |A| , |Az| / |A|   )

Solve:

$\left{5x+3y-5z=3\\-x-y+3z=1\\4x+y-z=4$  x =                           y =                         z =
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