Skip to main content
Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 5
Chapter 6, Problem 5

Solve each system in Exercises 5–18.
{x+y+2z=11x+y+3z=14x+2y−z=5\(\begin{cases}\)x + y + 2z = 11 \(\x\) + y + 3z = 14 \(\x\) + 2y - z = 5\(\end{cases}\)

Verified step by step guidance
1
Write down the system of equations clearly: \(\begin{cases} x + 0y + 2z = 11 \\ x + 0y + 3z = 14 \\ x + 2y - 0z = 5 \end{cases}\)
Notice that the first two equations both have \(x\) and \(z\) terms but no \(y\). Use these two equations to eliminate \(x\) or \(z\) by subtracting one equation from the other.
Subtract the first equation from the second: \( (x + 0y + 3z) - (x + 0y + 2z) = 14 - 11 \) which simplifies to an equation involving only \(z\).
Solve the resulting equation for \(z\). Once you have \(z\), substitute this value back into one of the first two equations to solve for \(x\).
With \(x\) and \(z\) known, substitute both into the third equation \(x + 2y = 5\) to solve for \(y\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
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 goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and set up these systems is essential for solving them.
Recommended video:
Guided course
4:27
Introduction to Systems of Linear Equations

Methods for Solving Systems

Common methods to solve systems include substitution, elimination, and matrix techniques like Gaussian elimination. Choosing an appropriate method depends on the system's structure. For example, elimination is useful when variables can be easily canceled by adding or subtracting equations.
Recommended video:
04:03
Choosing a Method to Solve Quadratics

Interpreting Coefficients and Variables

Coefficients represent the numerical multipliers of variables in equations. Recognizing zero coefficients helps simplify the system by reducing the number of variables in certain equations. This understanding aids in selecting the best approach to isolate variables and solve the system efficiently.
Recommended video:
Guided course
05:28
Equations with Two Variables
Related Practice
Textbook Question

Graph each inequality. y≤(1/3)x

2
views
Textbook Question

In Exercises 5–18, solve each system by the substitution method. {x+3y=8y=2x−9\(\begin{cases}\)x + 3y = 8 \(\y\) = 2x - 9\(\end{cases}\)

Textbook Question

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. x3+x2(x2+4)2\(\frac{x^3 + x^2}{(x^2 + 4)^2}\)

Textbook Question

In Exercises 5–18, solve each system by the substitution method. x + y = 4 y = 3x

1
views
Textbook Question

In Exercises 1–18, solve each system by the substitution method.{x2+y2=25x−y=1\(\begin{cases}\)x^2 + y^2 = 25 \(\x\) - y = 1\(\end{cases}\)

1
views
1
rank
Textbook Question

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. 5x2−6x+7(x−1)(x2+1)\(\frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)}\)