Skip to main content
Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 27
Chapter 6, Problem 27

Solve each system, using the method indicated.
3x + y = -7
x - y = -5 (Gaussian elimination)

Verified step by step guidance
1
Write the system of equations as an augmented matrix: \[\left[ \begin{array}{cc|c} 3 & 1 & -7 \\ 1 & -1 & -5 \end{array} \right]\]
Use row operations to create a leading 1 in the first row, first column if necessary. In this case, the second row already has a 1 in the first column, so swap Row 1 and Row 2 to make calculations easier.
Eliminate the \(x\)-term in the second row by replacing Row 2 with (Row 2) - 3*(Row 1). This will create a zero in the first column of the second row.
Solve for \(y\) from the second row after elimination, then substitute back into the first row to solve for \(x\).
Write the solution as an ordered pair \((x, y)\) representing the values that satisfy both equations.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to interpret and represent these systems is essential before applying any solving method.
Recommended video:
Guided course
4:27
Introduction to Systems of Linear Equations

Gaussian Elimination Method

Gaussian elimination is a systematic procedure to solve systems of linear equations by transforming the system's augmented matrix into row-echelon form using row operations. This method simplifies the system step-by-step, making it easier to solve for each variable through back substitution.
Recommended video:
Guided course
6:04
How to Multiply Equations in Elimination Method

Row Operations and Matrix Representation

Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. These operations are used on the augmented matrix of the system to simplify it without changing the solution set. Mastery of these operations is crucial for effective Gaussian elimination.
Recommended video:
Guided course
8:38
Performing Row Operations on Matrices
Related Practice
Textbook Question

Find each sum or difference, if possible.

[6−92413]+[−8256−34]\(\left\)[ \(\begin{matrix}\) 6 & -9 & 2 \\ 4 & 1 & 3 \(\end{matrix}\) \(\right\)] + \(\left\)[ \(\begin{matrix}\) -8 & 2 & 5 \\ 6 & -3 & 4 \(\end{matrix}\) \(\right\)]

Textbook Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

2x - y = 6

4x - 2y = 0

1
views
Textbook Question

Graph each inequality. y > 2x + 1

Textbook Question

Evaluate each determinant.

∣1−23000110−12∣\(\left\)| \(\begin{matrix}\) 1 & -2 & 3 \\ 0 & 0 & 0 \\ 1 & 10 & -12 \(\end{matrix}\) \(\right\)|

Textbook Question

Find the inverse, if it exists, for each matrix.[22−4260−3−35]\(\left\)[ \(\begin{matrix}\) 2 & 2 & -4 \\2 & 6 & 0 \\-3 & -3 & 5 \(\end{matrix}\) \(\right\)]

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. 1/(x(2x + 1)(3x2 + 4))