Solve each system in Exercises 25–26. (x+2)/6 − (y+4)/3 + z/2 = 0, (x+1)/2 + (y−1)/2 − z/4 = 9/2, (x−5)/4 + (y+1)/3 + (z−2)/2 = 19/4

Verified step by step guidance

1. Step 1: Start by examining the given system of equations. We have three equations with three variables: $x$, $y$, and $z$.
2. Step 2: Simplify each equation by eliminating the fractions. Multiply each term in the first equation by 6, the second by 4, and the third by 12 to clear the denominators.
3. Step 3: After clearing the fractions, rewrite the system of equations in standard form, aligning the variables and constants.
4. Step 4: Use the method of substitution or elimination to solve the system. You can start by isolating one variable in one of the equations and substituting it into the others.
5. Step 5: Continue solving the system by substituting back to find the values of the remaining variables. Check your solutions by substituting them back into the original equations to ensure they satisfy all three equations.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Solutions can be found using various methods, including substitution, elimination, or matrix operations.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. They can be represented in the form ax + by + cz = d, where a, b, c, and d are constants. Understanding how to manipulate and solve these equations is crucial for working with systems of equations.

Fractional Coefficients

In the given equations, coefficients are expressed as fractions, which can complicate calculations. It is important to understand how to work with fractions, including finding a common denominator and simplifying expressions, to effectively solve the system of equations.