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 13

Chapter 6, Problem 13

In Exercises 5–18, solve each system by the substitution method. 2x + 5y = - 4 3x - y = 11

Verified step by step guidance

Start with the given system of equations:

2x + 5 y = - 4 3 x - y = 11

Solve one of the equations for one variable in terms of the other. For example, solve the second equation for

y

3x - y = 11

becomes

y = 3x - 11

Substitute the expression for

y

from step 2 into the first equation: replace

y

2x + 5y = -4

with

3x - 11

to get

2x + 5(3x - 11) = -4

Simplify and solve the resulting equation for

x

. This will give you the value of

x

in terms of numbers.

Substitute the value of

x

back into the expression for

y

found in step 2 to find the corresponding value of

y

. This completes the solution to the system.

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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 variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to interpret and represent these systems is fundamental to solving them.

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 step-by-step.

Solving Linear Equations

Solving linear equations requires isolating the variable using inverse operations such as addition, subtraction, multiplication, or division. Mastery of these algebraic manipulations is essential to find the values of variables accurately.

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Graph each inequality. x²+y²≤1

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An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Textbook Question

Solve each system in Exercises 5–18. $\begin{cases}\)2x + y = 2 $\x$ + y - z = 4 \\3x + 2y + z = 0\(\end{cases}$

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\)xy = 3 $\x$^2 + y^2 = 10\(\end{cases}$