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 rewriting each equation in the system to simplify and express them in standard linear form. 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, first distribute the 4 on the left and the -3 on the right: $4(1+x)=-3(z-3y)$ becomes $4+4x=-3z+9y$.
4. Next, rearrange the third equation to standard form by moving all terms to one side: $4x-9y+3z=-4$.
5. Now, you have the system in standard form: 1) $6x\; +\; 3y\; +\; 5z\; =\; -1$ 2) $2x\; -\; 6y\; +\; 8z\; =\; -9$ 3) $4x\; -\; 9y\; +\; 3z\; =\; -4$ From here, you can use methods such as substitution, elimination, or matrix operations to solve for $x$, $y$, and $z$.

Key Concepts

Systems of Linear Equations

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

Simplifying and Rewriting Equations

Before solving, it is important to simplify each equation by distributing and combining like terms. This process transforms the system into a clearer form, making it easier to apply solution methods effectively.

Substitution and Elimination Methods

Substitution involves solving one equation for a variable and substituting into others, while elimination combines equations to eliminate a variable. Both methods reduce the system step-by-step to find the values of all variables.