Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

Previous problemNext problem

1:12 minutes

Problem 33

Textbook Question

In Exercises 33 - 36, write each matrix equation as a system of linear equations without matrices.

Verified step by step guidance

Identify the matrix equation: \( \begin{bmatrix} 4 & -7 \\ 2 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -3 \\ 1 \end{bmatrix} \).

Recognize that this represents a system of linear equations.

The first row of the matrix equation corresponds to the equation: \( 4x - 7y = -3 \).

The second row of the matrix equation corresponds to the equation: \( 2x - 3y = 1 \).

Write the system of equations: \( \begin{cases} 4x - 7y = -3 \\ 2x - 3y = 1 \end{cases} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Equation

A matrix equation is a mathematical expression that represents a system of linear equations in a compact form. It typically involves matrices and vectors, where the left side consists of a coefficient matrix multiplied by a variable vector, and the right side is a constant vector. Understanding how to interpret and manipulate these matrices is crucial for converting them into a system of equations.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed. They take the form ax + by = c, where a, b, and c are constants, and x and y are variables. In the context of a matrix equation, each row of the coefficient matrix corresponds to a linear equation, which can be extracted to form a system of equations.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. When converting a matrix equation into a system of equations, each equation represents a relationship between the variables, allowing for methods such as substitution or elimination to find solutions.