Linear Algebra – Linear Systems - Quick Guides

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Study GuideLinear Algebra–Linear Systems1.Gaussian EliminationWhat Is Gaussian Elimination?When we solve a system of linear equations, we are trying to find values for the variables that makeall equations true at the same time.Gaussian elimination is a step-by-step method that helps us do exactly that.The Main IdeaThe basic idea issimple:•Use one equation to eliminate a variable from another equation.•Keep doing this until only one variable is left.•Solve for that variable.•Substitute it back to find the remaining variables.This process of eliminating variables step by step is calledGaussian elimination.Example 1:Solving a SystemLet’s solve this system:Step 1:Eliminate One VariableWe want to eliminate (x).To do this, multiply the first equation by–3:

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Study GuideNow addthis to the second equation:When we add them:Great! The (x)-terms cancel out.Step 2:Solve for One VariableDivide both sides by–5:Step 3:Back-SubstitutionNow substitute (y = 1) into the first equation:So,Final AnswerThat’sit!

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Study GuideExample 2:Solving Using MatricesWe start with the same system:Step 1:Write the Coefficient MatrixWe first write just the coefficients of the variables:This is called thecoefficient matrix.Step 2:Create the Augmented MatrixNow we include the constants on the right side:This is called theaugmented matrix.Each row represents one equation:•First row → (x + y = 3)•Second row → (3x-2y = 4)Sometimes a vertical line is drawn toseparate coefficients from constants.Step 3:Perform Row OperationsInstead of changing equations, we now change rows.To eliminate (x) from the second row:

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Study GuideAdd–3 times the first rowto the second row.This gives:Now the second rowrepresents:So,Substitute back into the first row:Final solution:Same answer—but much more organized!Summary of Gaussian EliminationLet’s summarize the full method.If a system is written as:We form theaugmented matrix:Then we performelementary row operations.

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Study GuideElementary Row OperationsThere are only three types:1.Interchange Two RowsSwap any two rows.2.Multiply a Row by a Nonzero NumberMultiply every entry in a row by the same nonzero constant.3.Add a Multiple of One Row to Another RowThis is the most common operation.It helps eliminate variables.Goal of Row OperationsWe want to transform the matrix into a simpler form calledechelon form.A matrix is in echelon form if:•All zero rows are at thebottom.•The first nonzero number in each row (called a pivot) is to the right of the pivot above it.•Entries below each pivot are zero.In this form, the system becomes much easier to solve usingback-substitution.Important ConceptRow operationsdo not change the solutionof the system.That means:If the simpler matrix representsThen its solutions are exactly the same as the original system:

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Study GuideFinal ThoughtsGaussian elimination is powerful because:•It works for small and largesystems.•It reduces messy algebra.•It provides a systematic, reliable method.Once you understand the logic—eliminate, simplify, back-substitute—solving systems becomesmuch easier and more structured.Example 3:Solving a 3-Variable System Using Gaussian EliminationThe System of EquationsWe are given:Our goal is to find the values ofx, y, and zthat make all three equations true at the same time.Step 1:Write the Augmented MatrixTo organize the system, we write the augmentedmatrix:Each row represents one equation.The last column contains the constants on the right side.

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Study GuideForward EliminationOur goal now is to create zerosbelow each pivot(the first nonzero number in a row).Step 2:Eliminate the First Variable (x)We want zeros below the first entry in the first column.To eliminate (x) from the second and third rows:•Add–2 times row 1to row 2•Add–4 times row 1to row 3After doing this, the matrix becomes:Great! Now the first column has zeros below thepivot.Step 3:Eliminate the Second Variable (y)Now we focus on the second column.We want a zero below the pivot in row 2.One way would be to add a fraction of row 2 to row 3.But to avoid fractions, we firstswap row 2 and row 3.Afterswitching rows 2 and 3:Swapping rows doesnotchange the solution of the system.Step 4:Eliminate y from Row 3

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Study GuideNow subtract5 times row 2from row 3:This gives:Now the matrix is inechelon form.All entries below each pivot are zero.The forward elimination part is complete!Back SubstitutionNow we solve from the bottom up.Step 5:Solve for zThe third row represents:So,Step 6:Solve for ySubstitute (z = 1) into the second row:

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Study GuideStep 7:Solve for xSubstitute (y = 2) and (z = 1)into the first row:Final AnswerExample 4:Solving a System Using Gaussian EliminationThe System of EquationsWe are given:

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Study GuideOur goal is to find the values ofx, y, and z.Step 1:Write the Augmented MatrixTo organize the system, wewrite:Each row represents one equation.The last column contains the constants.Forward EliminationOur goal is to create zeros below each pivot.Step 2:Simplify the First RowTo make calculations easier, multiplyRow 1 by 1/2:This gives:Now the first pivot is 1. Much cleaner!

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