Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 15

Chapter 6, Problem 15

In Exercises 5–18, solve each system by the substitution method. 2x - 3y = 8 - 2x 3x + 4y = x + 3y + 14

Verified step by step guidance

Step 1: Simplify each equation to standard form. For the first equation, combine like terms:

2x - 3 y = 8 - 2 x

becomes

2x + 2x - 3 y = 8

, which simplifies to

4 x - 3 y = 8

Step 2: Simplify the second equation similarly:

3 x + 4 y = x + 3 y + 14

becomes

3 x - x + 4 y - 3 y = 14

, which simplifies to

2 x + y = 14

Step 3: Solve one of the equations for one variable. For example, solve the second equation for

y

y = 14 - 2 x

Step 4: Substitute the expression for

y

from Step 3 into the first equation:

4 x - 3 (14 - 2 x) = 8

Step 5: Simplify and solve the resulting equation for

x

. Then, substitute the value of

x

back into the expression for

y

to find the value of

y

Verified video answer for a similar problem:

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

Video duration:

Was this helpful?

Key Concepts

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

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to interpret and manipulate these equations is essential for solving the system.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is particularly useful when one variable is easily isolated.

Simplifying and Rearranging Equations

Before applying substitution, equations often need to be simplified and rearranged to isolate variables or combine like terms. This step ensures clarity and accuracy in substitution and helps avoid errors during solving. Mastery of algebraic manipulation is crucial here.

Recommended video:

Guided course

5:48

Adding & Subtracting Unlike Radicals by Simplifying

Related Practice

Textbook Question

Graph each inequality. x²+y²≤1

views

Textbook Question

Find the quadratic function y = ax^2 + bx + c whose graph passes through the points (1, 4), (3, 20), and (-2, 25).

views

Textbook Question

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\)x + y = 1 $\x$^2 + xy - y^2 = -5\(\end{cases}$

Textbook Question

Write the partial fraction decomposition of each rational expression. 9x+21/(x² + 2x - 15)

Textbook Question

Write the partial fraction decomposition of each rational expression. 4/(2x² -5x -3)

Textbook Question

In Exercises 1–26, graph each inequality. x²+y²>25

views