Math Problem Statement
Write an augmented matrix for the following system of equations.
negative 2 x−2x
plus+
8 y8y
equals=
66
3 x3x
minus−
5 y5y
equals=
88
Question content area bottom
Part 1
The entries in the matrix are:
left bracket Start 2 By 3 Table 1st Row 1st Column nothing 2nd Column nothing 3rd Column nothing 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing EndTable right bracket
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Solution
To write the augmented matrix for the given system of equations, we first rewrite the system clearly:
The augmented matrix takes the form:
\text{coefficients of $$x$$ and $$y$$ from equation 1} & \text{constant from equation 1} \\ \text{coefficients of $$x$$ and $$y$$ from equation 2} & \text{constant from equation 2} \\ \end{bmatrix}$$ ### Step-by-Step Solution: 1. From the first equation: coefficients are $$-2$$ and $$8$$, and the constant is $$6$$. 2. From the second equation: coefficients are $$3$$ and $$-5$$, and the constant is $$8$$. The augmented matrix is: $$\begin{bmatrix} -2 & 8 & 6 \\ 3 & -5 & 8 \\ \end{bmatrix}$$ ### Final Answer: $$\begin{bmatrix} -2 & 8 & 6 \\ 3 & -5 & 8 \\ \end{bmatrix}$$ Would you like a detailed explanation of how to solve the system using this matrix? --- #### 5 Related Questions: 1. How do you solve a system of linear equations using Gaussian elimination? 2. What is the significance of augmented matrices in solving systems of equations? 3. Can this system be solved using the inverse matrix method? If so, how? 4. How do row operations affect the solution of an augmented matrix? 5. What are the conditions for a system of linear equations to have a unique solution? #### Tip: Always double-check that the coefficients in your augmented matrix align with the variables in the equations to avoid mistakes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Matrix Representation
Formulas
Augmented matrix representation: coefficients of variables form the left side, constants form the right side.
Theorems
Matrix representation of linear systems
Row equivalence of matrices
Suitable Grade Level
Grades 8-10
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