Solve each system in Exercises 5–18. 3(2x+y)+5z=−1, 2(x−3y+4z)=−9, 4(1+x)=−3(z−3y)

Verified step by step guidance

1. Start by expanding each equation to simplify them. For the first equation, distribute the 3: $3(2x + y) + 5z = -1$ becomes $6x + 3y + 5z = -1$.
2. For the second equation, distribute the 2: $2(x - 3y + 4z) = -9$ becomes $2x - 6y + 8z = -9$.
3. For the third equation, distribute the 4 on the left side and the -3 on the right side: $4(1 + x) = -3(z - 3y)$ becomes $4 + 4x = -3z + 9y$.
4. Rearrange the third equation to isolate terms: $4x - 9y + 3z = -4$.
5. Now, you have a system of three linear equations: $6x + 3y + 5z = -1$, $2x - 6y + 8z = -9$, and $4x - 9y + 3z = -4$. Use methods such as substitution or elimination to solve for $x$, $y$, and $z$.

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 mathematical statements that express a relationship between variables in a straight-line format, typically written as Ax + By + Cz = D. Each equation in a system can represent a geometric plane in three-dimensional space, and the solution to the system corresponds to the intersection of these planes.

Substitution and Elimination Methods

Substitution and elimination are two common techniques for solving systems of equations. The substitution method involves solving one equation for a variable and substituting that expression into the other equations. The elimination method involves adding or subtracting equations to eliminate a variable, simplifying the system to make it easier to solve.