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Math  /  Algebra

QuestionExpress the following set of linear equations in matrix form: 2x1+4x2−5x3=−7x1−3x2+x3=103x1+5x2+3x3=2\begin{aligned} 2 x_{1}+4 x_{2}-5 x_{3} & =-7 \\ x_{1}-3 x_{2}+x_{3} & =10 \\ 3 x_{1}+5 x_{2}+3 x_{3} & =2 \end{aligned}

Studdy Solution

STEP 1

1. We are given a system of linear equations.
2. The goal is to express these equations in matrix form, which consists of a coefficient matrix, a variable vector, and a constant vector.

STEP 2

1. Identify the coefficient matrix from the given equations.
2. Identify the variable vector.
3. Identify the constant vector.
4. Combine these components to express the system in matrix form.

STEP 3

Identify the coefficients of the variables x1x_1, x2x_2, and x3x_3 from each equation to form the coefficient matrix:
The coefficients are: - First equation: 2,4,−52, 4, -5 - Second equation: 1,−3,11, -3, 1 - Third equation: 3,5,33, 5, 3
Thus, the coefficient matrix is:
[24−51−31353]\begin{bmatrix} 2 & 4 & -5 \\ 1 & -3 & 1 \\ 3 & 5 & 3 \end{bmatrix}

STEP 4

Identify the variable vector, which consists of the variables x1x_1, x2x_2, and x3x_3:
[x1x2x3]\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}

STEP 5

Identify the constant vector from the right-hand side of each equation:
The constants are: - First equation: −7-7 - Second equation: 1010 - Third equation: 22
Thus, the constant vector is:
[−7102]\begin{bmatrix} -7 \\ 10 \\ 2 \end{bmatrix}

STEP 6

Combine the coefficient matrix, variable vector, and constant vector to express the system in matrix form:
[24−51−31353][x1x2x3]=[−7102]\begin{bmatrix} 2 & 4 & -5 \\ 1 & -3 & 1 \\ 3 & 5 & 3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -7 \\ 10 \\ 2 \end{bmatrix}

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