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

Key Concepts

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously. Methods for solving systems include substitution, elimination, and matrix operations. Understanding how to manipulate and solve these equations is crucial for finding the values of the variables involved.

Linear Equations

Linear equations are mathematical statements that express a relationship between variables in a straight-line format, typically written as Ax + By + Cz = D. Each equation in the system represents a plane in three-dimensional space, and the solution is the point where these planes intersect. Recognizing the structure of linear equations helps in applying appropriate solving techniques.

Fractional Coefficients

Fractional coefficients occur when the variables in an equation are multiplied by fractions, which can complicate calculations. To solve equations with fractional coefficients, it is often helpful to eliminate the fractions by multiplying through by the least common denominator. This simplification makes it easier to manipulate the equations and find the solution to the system.