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}⎩⎨⎧​6x+2​−3y+4​+2z​=02x+1​+2y−1​−4z​=29​4x−5​+3y+1​+2z−2​=419​​

Accepted Answer

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.

Solve each system in Exercises 25–26. $\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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Key Concepts

Systems of Linear Equations

Clearing Fractions and Simplifying Equations

Methods for Solving Systems: Substitution and Elimination