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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 25
Chapter 6, Problem 25

In Exercises 19–28, solve each system by the addition method. {x2+y2=25(x−8)2+y2=41\(\begin{cases}\)x^2 + y^2 = 25 \\(x - 8)^2 + y^2 = 41\(\end{cases}\)

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Identify the system of equations: \(x^2 + y^2 = 25\) and \((x - 8)^2 + y^2 = 41\).
Expand the second equation: \((x - 8)^2 + y^2 = x^2 - 16x + 64 + y^2 = 41\).
Rewrite the expanded second equation as \(x^2 - 16x + 64 + y^2 = 41\).
Subtract the first equation \(x^2 + y^2 = 25\) from the expanded second equation to eliminate \(y^2\) and \(x^2\): \((x^2 - 16x + 64 + y^2) - (x^2 + y^2) = 41 - 25\).
Simplify the resulting equation to isolate \(x\) and solve for its value(s).

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Key Concepts

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

Systems 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. In this problem, the system involves two equations representing circles, and solving the system means finding their points of intersection.
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Introduction to Systems of Linear Equations

Addition Method (Elimination Method)

The addition method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. This method is typically used for linear systems, but can be adapted here by expanding and simplifying the given equations to linear form before elimination.
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Choosing a Method to Solve Quadratics

Equations of Circles and Their Expansion

Each equation represents a circle in standard form. Expanding the squared terms converts the equations into polynomial form, which can then be manipulated algebraically. Understanding how to expand and simplify these equations is essential to apply the addition method effectively.
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Circles in Standard Form
Related Practice
Textbook Question

Solve each system in Exercises 25–26. {x+26−y+43+z2=0x+12+y−12−z4=92x−54+y+13+z−22=194\(\begin{cases}\[\frac{x + 2}{6}\) - \(\frac{y + 4}{3}\) + \(\frac{z}{2}\) = 0 \(\frac{x + 1}{2}\) + \(\frac{y - 1}{2}\) - \(\frac{z}{4}\) = \(\frac{9}{2}\) \(\frac{x - 5}{4}\) + \(\frac{y + 1}{3}\) + \(\frac{z - 2}{2}\) = \(\frac{19}{4}\]\end{cases}\)

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Textbook Question

Graph each inequality. y≥log2(x+1)

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Textbook Question

In Exercises 19–30, solve each system by the addition method. 4x + 3y = 15 2x - 5y = 1

Textbook Question

Write the partial fraction decomposition of each rational expression. x2+2x+7/x(x − 1)2

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Textbook Question

Solve each system by the method of your choice.

{5y=x2−1x−y=1\(\begin{cases}\) 5y = x^2 - 1 \\ x - y = 1 \(\end{cases}\)

Textbook Question

Solve each system in Exercises 25–26. {x+32−y−12+z+24=32x−52+y+13−z4=−256x−34−y+12+z−32=−52\(\begin{cases}\[\frac{x + 3}{2}\) - \(\frac{y - 1}{2}\) + \(\frac{z + 2}{4}\) = \(\frac{3}{2}\) \(\frac{x - 5}{2}\) + \(\frac{y + 1}{3}\) - \(\frac{z}{4}\) = - \(\frac{25}{6}\) \(\frac{x - 3}{4}\) - \(\frac{y + 1}{2}\) + \(\frac{z - 3}{2}\) = - \(\frac{5}{2}\]\end{cases}\)

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