Solve each system in Exercises 25–26. (x+3)/2 − (y−1)/2 + (z+2)/4 = 3/2, (x−5)/2 + (y+1)/3 − z/4 = − 25/6, (x−3)/4 − (y+1)/2 + (z−3)/2= − 5/2

Verified step by step guidance

1. Start by rewriting each equation to clear the denominators. Multiply both sides of each equation by the least common denominator (LCD) of the fractions in that equation to eliminate the fractions.
2. After clearing denominators, simplify each equation to get a linear equation in terms of x, y, and z. This will give you a system of three linear equations without fractions.
3. Organize the system of equations in standard form: $ax+by+cz=d$ for each equation, where a, b, c, and d are constants.
4. Use either substitution, elimination, or matrix methods (such as Gaussian elimination) to solve the system. For example, you can solve one equation for one variable and substitute into the others, or combine equations to eliminate variables step-by-step.
5. Continue solving until you find expressions for x, y, and z. Verify your solution by substituting the values back into the original equations to ensure they satisfy all three.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and set up these systems is essential for solving them.

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 and prepares it for methods like substitution or elimination.

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

Common techniques to solve systems include substitution (solving one equation for a variable and substituting), elimination (adding or subtracting equations to eliminate a variable), and matrix methods like Gaussian elimination. Choosing the right method depends on the system's complexity.