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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 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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1
Start with the given system of equations: \(x^2 + y^2 = 25\) and \(x - y = 1\).
From the second equation, solve for one variable in terms of the other. For example, express \(x\) as \(x = y + 1\).
Substitute the expression for \(x\) into the first equation to replace \(x\) with \(y + 1\), resulting in \((y + 1)^2 + y^2 = 25\).
Expand the squared term and combine like terms to form a quadratic equation in terms of \(y\): \(y^2 + 2y + 1 + y^2 = 25\).
Simplify the equation and solve the quadratic for \(y\). Then, use the values of \(y\) to find the corresponding \(x\) values using \(x = y + 1\).

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Key Concepts

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

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 or easily solvable for one variable.
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Solving Quadratic Equations

When substitution leads to a quadratic equation, you must solve it using methods such as factoring, completing the square, or the quadratic formula. Quadratic equations can have two, one, or no real solutions, which correspond to the number of intersection points between the curves represented by the system.
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Systems of Nonlinear Equations

A system involving equations like x² + y² = 25 (a circle) and x - y = 1 (a line) is nonlinear. Understanding the geometric interpretation helps: solutions correspond to intersection points of the circle and line. Such systems may have zero, one, or two solutions depending on how the line intersects the circle.
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Related Practice
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\(\end{cases}\)

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Textbook Question

Graph each inequality. y>2x−1

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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\(\frac{x^3 + x^2}{(x^2 + 4)^2}\)

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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

Solve each system in Exercises 5–18. {4x−y+2z=11x+2y−z=−12x+2y−3z=−1\(\begin{cases}\)4x - y + 2z = 11 \(\x\) + 2y - z = -1 \\2x + 2y - 3z = -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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