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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 5 - Systems of Equations and Inequalities문제 25
6장, 문제 25

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 \\ \(\frac{x + 1}{2}\) + \(\frac{y - 1}{2}\) - \(\frac{z}{4}\) = \(\frac{9}{2}\) \\ \(\frac{x - 5}{4}\) + \(\frac{y + 1}{3}\) + \(\frac{z - 2}{2}\) = \(\frac{19}{4}\) \(\end{cases}\)

검증된 단계별 안내
1
Start by rewriting each equation to clear the denominators. Multiply both sides of each equation by the least common multiple (LCM) of the denominators to eliminate fractions. For example, for the first equation, multiply through by 6, for the second by 4, and for the third by 12.
After clearing denominators, simplify each equation by distributing and combining like terms. This will give you a system of three linear equations in standard form: $Ax + By + Cz = D$.
Organize the system of equations clearly, aligning the variables \(x\), \(y\), and \(z\) on the left side and constants on the right side. This will help in applying methods such as substitution, elimination, or matrix operations.
Choose a method to solve the system: substitution, elimination, or using matrices (such as Gaussian elimination). For substitution or elimination, solve one equation for one variable and substitute into the others to reduce the system step-by-step.
Continue simplifying and substituting until you find the values of \(x\), \(y\), and \(z\). Verify your solution by plugging the values back into the original equations to ensure all are satisfied.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Systems 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 manipulate these systems is essential for finding the common solution point.
추천 영상:
4:27
Introduction to Systems of Linear Equations

Clearing Fractions and Simplifying Equations

Equations with fractions can be simplified by multiplying both sides by the least common denominator to eliminate fractions. This step makes the system easier to work with by converting it into standard linear form, facilitating methods like substitution or elimination.
추천 영상:
05:45
Radical Expressions with Fractions

Methods for Solving Systems: Substitution and Elimination

Common techniques to solve systems include substitution, where one variable is expressed in terms of others, and elimination, where equations are added or subtracted to eliminate a variable. Mastery of these methods allows efficient solving of multi-variable linear systems.
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5:48
Solving Systems of Equations - Substitution
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