Math Problem Statement

Determine the solution sets for each of the following systems of equations using the gaussian elimination method 1:24x-15y=30 -8x+5y=-20 2:5x-3y=17 -2x+5y=-22 3: -x+2y=-8 3x-6y=24 4:8x-3y=6 3x+5y=-10 5:12x-6y=21 -4x+2y=-7 6:x-y=0 3x+4y=-21

Solution

We are tasked with solving six systems of linear equations using Gaussian Elimination. The method involves performing row operations to reduce the system of equations to an upper triangular matrix and then applying back substitution to find the solution.

Let's go through the systems one by one.

1. System of Equations

24x - 15y &= 30 \quad (1) \\ -8x + 5y &= -20 \quad (2) \end{aligned}$$ **Step 1: Form the augmented matrix:** $$\begin{pmatrix} 24 & -15 & | & 30 \\ -8 & 5 & | & -20 \end{pmatrix}$$ **Step 2: Eliminate the $$x$$-term in the second equation.** We can divide the first row by 24 and the second row by -8. $$\begin{pmatrix} 1 & -\frac{5}{8} & | & \frac{5}{4} \\ 1 & -\frac{5}{8} & | & \frac{5}{4} \end{pmatrix}$$ --- ### 2. System of Equations $$\begin{aligned} 5x - 3y &= 17 \quad (1) \\ -2x + 5y &= -22 \quad (2) \end{aligned}$$ --- ### 3. System of Equations $$\begin{aligned} -x + 2y &= -8 \quad (1) \\ 3x - 6y &= 24 \quad (2) \end{aligned}$$ --- ### 4. System of Equations $$\begin{aligned} 8x - 3y &= 6 \quad (1) \\ 3x + 5y &= -10 \quad (2) \end{aligned}$$ --- ### 5. System of Equations $$\begin{aligned} 12x - 6y &= 21 \quad (1) \\ -4x + 2y &= -7 \quad (2) \end{aligned}$$ --- ### 6. System of Equations $$\begin{aligned} x - y &= 0 \quad (1) \\ 3x + 4y &= -21 \quad (2) \end{aligned}$$ ---

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Gaussian Elimination

Formulas

Row operations for Gaussian elimination
Back substitution

Theorems

Gaussian elimination method

Suitable Grade Level

Grades 9-12

Related Recommendation

Solving System of Linear Equations with Gaussian Elimination

Solve a System of Linear Equations Using Gaussian Elimination

Solve System of Equations with Gaussian Elimination - Step-by-Step Guide

Solving a System of Equations Using Gaussian Elimination

Solving Systems of Linear Equations Using Gauss and Gauss-Jordan Methods