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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 7
Chapter 6, Problem 7

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

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Identify the system of equations: \(\begin{cases} x + 3y = 8 \\ y = 2x - 9 \end{cases}\).
Since the second equation is already solved for \(y\), substitute \(y = 2x - 9\) into the first equation.
Replace \(y\) in the first equation: \(x + 3(2x - 9) = 8\).
Simplify and solve the resulting equation for \(x\): \(x + 6x - 27 = 8\).
Once you find \(x\), substitute it back into \(y = 2x - 9\) to find the value of \(y\).

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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 set of 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.
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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 especially useful when one equation is already solved for a variable.
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Solving Linear Equations

Solving linear equations means finding the value(s) of the variable(s) that make the equation true. This often involves isolating the variable using inverse operations like addition, subtraction, multiplication, or division. Mastery of these techniques is essential for solving systems after substitution.
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Related Practice
Textbook Question

Graph each inequality. y≤(1/3)x

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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

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Textbook Question

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\(\end{cases}\)

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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\(\end{cases}\)

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Textbook Question

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.

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