Solve by eliminating variables:

Verified step by step guidance

1. Step 1: Write down the system of equations clearly: $\begin{array}{}x-6y=-222x+4y-3z=293x-2y+5z=-17\end{array}$
2. Step 2: Choose two pairs of equations to eliminate the same variable. For example, eliminate $x$ first by manipulating the first and second equations, and then the first and third equations.
3. Step 3: To eliminate $x$ between the first and second equations, multiply the first equation by 2 (to match the coefficient of $x$ in the second equation), then subtract the second equation from this result.
4. Step 4: Similarly, to eliminate $x$ between the first and third equations, multiply the first equation by 3 (to match the coefficient of $x$ in the third equation), then subtract the third equation from this result.
5. Step 5: After these operations, you will have two new equations in terms of $y$ and $z$. Solve this new system using elimination or substitution to find the values of $y$ and $z$. Then substitute back into one of the original equations to find $x$.

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 manipulate these systems is fundamental in algebra.

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, simplifying the system step-by-step. This technique helps reduce the number of variables, making it easier to solve for the remaining unknowns.

Manipulating Equations with Multiple Variables

When dealing with three variables, it is important to strategically multiply or combine equations to align coefficients for elimination. Careful manipulation ensures variables are eliminated correctly, leading to a solvable system of two equations with two variables.