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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 17
Chapter 6, Problem 17

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

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Start with the given system of equations: \(x + y = 1\) and \((x - 1)^2 + (y + 2)^2 = 10\).
From the first equation, solve for one variable in terms of the other. For example, express \(y\) as \(y = 1 - x\).
Substitute the expression for \(y\) into the second equation to replace \(y\) with \(1 - x\). This gives: \((x - 1)^2 + ((1 - x) + 2)^2 = 10\).
Simplify the equation by expanding the squares and combining like terms to form a quadratic equation in terms of \(x\).
Solve the quadratic equation for \(x\), then substitute each solution back into \(y = 1 - x\) to find the corresponding \(y\) values.

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Key Concepts

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

System of Equations

A system of equations consists of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and manipulate these systems is essential for finding their solutions.
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Introduction to Systems of Linear Equations

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 solved for a variable.
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Solving Quadratic Equations

When substitution leads to a quadratic equation, techniques such as factoring, completing the square, or using the quadratic formula are used to find solutions. Recognizing and solving quadratic equations is crucial for systems involving nonlinear equations like circles or parabolas.
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Related Practice
Textbook Question

In Exercises 19–28, solve each system by the addition method. {x2+y2=13x2−y2=5\(\begin{cases}\)x^2 + y^2 = 13 \(\x\)^2 - y^2 = 5\(\end{cases}\)

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Textbook Question

Solve each system in Exercises 5–18. {3(2x+y)+5z=−12(x−3y+4z)=−94(1+x)=−3(z−3y)\(\begin{cases}\)3(2x + y) + 5z = -1 \\2(x - 3y + 4z) = -9 \\4(1 + x) = -3(z - 3y)\(\end{cases}\)

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Textbook Question

Write the partial fraction decomposition of each rational expression. 4x2+13x-9/x (x − 1)(x+3)

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Textbook Question

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^2 - 3x - 4)/x(x + 2)(x - 1)

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Textbook Question

In Exercises 5–18, solve each system by the substitution method. y = (1/3)x + 2/3 y = (5/7)x - 2

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Textbook Question

Graph each inequality. y < x2 - 1

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