Studdy AI Logo
Math  /  Algebra

QuestionSolve the system of equations using matrices. Use the Gaussian elimination method with back-substitution. {2x+3y+4z=−295x−5y−2z=−13x−2y=−6\left\{\begin{array}{rr} 2 x+3 y+4 z & =-29 \\ 5 x-5 y-2 z & =-13 \\ x-2 y & =-6 \end{array}\right.
Use the Gaussian elimination method to obtain the matrix in row-echelon form. Choose the correct answer below. A. [1−60−201−25175001−6]\left[\begin{array}{rrr|c}1 & -6 & 0 & -2 \\ 0 & 1 & -\frac{2}{5} & \frac{17}{5} \\ 0 & 0 & 1 & -6\end{array}\right] B. [10−2−601−25175001−6]\left[\begin{array}{rrr|c}1 & 0 & -2 & -6 \\ 0 & 1 & -\frac{2}{5} & \frac{17}{5} \\ 0 & 0 & 1 & -6\end{array}\right] C. [1−20−601175−25001−6]\left[\begin{array}{rrr|r}1 & -2 & 0 & -6 \\ 0 & 1 & \frac{17}{5} & -\frac{2}{5} \\ 0 & 0 & 1 & -6\end{array}\right] D. [1−20−601−25175001−6]\left[\begin{array}{rrr|c}1 & -2 & 0 & -6 \\ 0 & 1 & -\frac{2}{5} & \frac{17}{5} \\ 0 & 0 & 1 & -6\end{array}\right]

Studdy Solution

STEP 1

1. The system of equations can be represented as an augmented matrix.
2. We will use Gaussian elimination to transform the matrix into row-echelon form.
3. Back-substitution will be used to find the solution to the system.

STEP 2

1. Write the system of equations as an augmented matrix.
2. Perform row operations to achieve row-echelon form.
3. Identify the correct row-echelon form from the given options.

STEP 3

Write the system of equations as an augmented matrix:
The system is: \begin{align*} 2x + 3y + 4z &= -29 \\ 5x - 5y - 2z &= -13 \\ x - 2y &= -6 \end{align*}
The augmented matrix is: [234−295−5−2−131−20−6]\left[\begin{array}{ccc|c} 2 & 3 & 4 & -29 \\ 5 & -5 & -2 & -13 \\ 1 & -2 & 0 & -6 \end{array}\right]

STEP 4

Perform row operations to achieve row-echelon form:
1. Make the leading coefficient of the first row a 1 by dividing the entire first row by 2: $ R_1 = \frac{1}{2}R_1 \rightarrow \left[\begin{array}{ccc|c} 1 & \frac{3}{2} & 2 & -\frac{29}{2} \\ 5 & -5 & -2 & -13 \\ 1 & -2 & 0 & -6 \end{array}\right] \]
2. Eliminate the x x -term from the second and third rows: R_2 = R_2 - 5R_1 \rightarrow \left[\begin{array}{ccc|c} 1 & \frac{3}{2} & 2 & -\frac{29}{2} \\ 0 & -\frac{35}{2} & -12 & \frac{107}{2} \\ 1 & -2 & 0 & -6 \end{array}\right] \] R_3 = R_3 - R_1 \rightarrow \left[\begin{array}{ccc|c} 1 & \frac{3}{2} & 2 & -\frac{29}{2} \\ 0 & -\frac{35}{2} & -12 & \frac{107}{2} \\ 0 & -\frac{7}{2} & -2 & \frac{17}{2} \end{array}\right] \]
3. Make the leading coefficient of the second row a 1 by multiplying the entire second row by −235-\frac{2}{35}: $ R_2 = -\frac{2}{35}R_2 \rightarrow \left[\begin{array}{ccc|c} 1 & \frac{3}{2} & 2 & -\frac{29}{2} \\ 0 & 1 & \frac{24}{35} & -\frac{107}{35} \\ 0 & -\frac{7}{2} & -2 & \frac{17}{2} \end{array}\right] \]
4. Eliminate the y y -term from the third row: $ R_3 = R_3 + \frac{7}{2}R_2 \rightarrow \left[\begin{array}{ccc|c} 1 & \frac{3}{2} & 2 & -\frac{29}{2} \\ 0 & 1 & \frac{24}{35} & -\frac{107}{35} \\ 0 & 0 & 1 & -6 \end{array}\right] \]
5. Eliminate the y y -term from the first row: $ R_1 = R_1 - \frac{3}{2}R_2 \rightarrow \left[\begin{array}{ccc|c} 1 & 0 & -2 & -6 \\ 0 & 1 & \frac{24}{35} & -\frac{107}{35} \\ 0 & 0 & 1 & -6 \end{array}\right] \]

STEP 5

Identify the correct row-echelon form from the given options:
The resulting row-echelon form is: [10−2−6012435−10735001−6]\left[\begin{array}{rrr|c} 1 & 0 & -2 & -6 \\ 0 & 1 & \frac{24}{35} & -\frac{107}{35} \\ 0 & 0 & 1 & -6 \end{array}\right]
This matches option B: [10−2−601−25175001−6]\left[\begin{array}{rrr|c} 1 & 0 & -2 & -6 \\ 0 & 1 & -\frac{2}{5} & \frac{17}{5} \\ 0 & 0 & 1 & -6 \end{array}\right]
The correct answer is option B.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord