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Linear Algebra – Linear Systems

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    Study Guide Linear Algebra – Linear Systems 1. Gaussian Elimination What Is Gaussian Elimination? When we solve a system of linear equations, we are trying to find values for the variables that make all equations true at the same time . Gaussian elimination is a step - by - step method that helps us do exactly that. The Main Idea The basic idea is simple: • 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 called Gaussian elimination . Example 1: Solving a System Let’s solve this system: Step 1: Eliminate One Variable We want to eliminate (x). To do this, multiply the first equation by – 3 :

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    Study Guide Now add this to the second equation: When we add them: Great! The (x) - terms cancel out. Step 2: Solve for One Variable Divide both sides by – 5: Step 3: Back - Substitution Now substitute (y = 1) into the first equation: So, Final Answer That’s it!

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    Study Guide Example 2: Solving Using Matrices We start with the same system: Step 1: Write the Coefficient Matrix We first write just the coefficients of the variables: This is called the coefficient matrix . Step 2: Create the Augmented Matrix Now we include the constants on the right side: This is called the augmented matrix . Each row represents one equation: • First row → (x + y = 3) • Second row → (3x - 2y = 4) Sometimes a vertical line is drawn to separate coefficients from constants. Step 3: Perform Row Operations Instead of changing equations, we now change rows. To eliminate (x) from the second row:

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    Study Guide Add – 3 times the first row to the second row. This gives: Now the second row represents: So, Substitute back into the first row: Final solution: Same answer — but much more organized! Summary of Gaussian Elimination Let’s summarize the full method. If a system is written as: We form the augmented matrix : Then we perform elementary row operations .

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    Study Guide Elementary Row Operations There are only three types: 1. Interchange Two Rows Swap any two rows. 2. Multiply a Row by a Nonzero Number Multiply every entry in a row by the same nonzero constant. 3. Add a Multiple of One Row to Another Row This is the most common operation. It helps eliminate variables. Goal of Row Operations We want to transform the matrix into a simpler form called echelon form . A matrix is in echelon form if: • All zero rows are at the bottom. • 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 using back - substitution . Important Concept Row operations do not change the solution of the system. That means: If the simpler matrix represents Then its solutions are exactly the same as the original system:

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    Study Guide Final Thoughts Gaussian elimination is powerful because: • It works for small and large systems. • It reduces messy algebra. • It provides a systematic, reliable method. Once you understand the logic — eliminate, simplify, back - substitute — solving systems becomes much easier and more structured. Example 3: Solving a 3 - Variable System Using Gaussian Elimination The System of Equations We are given: Our goal is to find the values of x, y, and z that make all three equations true at the same time. Step 1: Write the Augmented Matrix To organize the system, we write the augmented matrix: Each row represents one equation. The last column contains the constants on the right side.

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    Study Guide Forward Elimination Our goal now is to create zeros below 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 1 to row 2 • Add – 4 times row 1 to row 3 After doing this, the matrix becomes: Great! Now the first column has zeros below the pivot. 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 first swap row 2 and row 3 . After switching rows 2 and 3: Swapping rows does not change the solution of the system. Step 4: Eliminate y from Row 3

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    Study Guide Now subtract 5 times row 2 from row 3: This gives: Now the matrix is in echelon form . All entries below each pivot are zero. The forward elimination part is complete! Back Substitution Now we solve from the bottom up. Step 5: Solve for z The third row represents: So, Step 6: Solve for y Substitute (z = 1) into the second row:

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    Study Guide Step 7: Solve for x Substitute (y = 2) and (z = 1) into the first row: Final Answer Example 4: Solving a System Using Gaussian Elimination The System of Equations We are given:

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    Study Guide Our goal is to find the values of x, y, and z . Step 1: Write the Augmented Matrix To organize the system, we write: Each row represents one equation. The last column contains the constants. Forward Elimination Our goal is to create zeros below each pivot. Step 2: Simplify the First Row To make calculations easier, multiply Row 1 by 1/2 : This gives: Now the first pivot is 1. Much cleaner!

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