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}⎩⎨⎧​2x+3​−2y−1​+4z+2​=23​2x−5​+3y+1​−4z​=−625​4x−3​−2y+1​+2z−3​=−25​​

Accepted Answer

First, rewrite each equation to eliminate the denominators by multiplying through by the least common multiple (LCM) of the denominators in each equation. This will give you equations without fractions, making them easier to work with. Next, simplify each resulting equation by distributing and combining like terms to get a linear equation in terms of x, y, and z. After simplifying, write the system of three linear equations clearly, each in the form $$Ax + By + Cz = D. $$Use either the substitution method, elimination method, or matrix methods (such as Gaussian elimination) to solve the system step-by-step. For example, you can solve one equation for one variable and substitute into the others, or eliminate variables by adding or subtracting equations. Continue the process until you find the values of x, y, and z that satisfy all three equations simultaneously.

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Key Concepts

Systems of Linear Equations

Clearing Fractions and Simplifying Equations

Methods for Solving Systems: Substitution, Elimination, and Matrix Methods