Solve each system in Exercises 5–18. x+0y+2z=11, x+0y+3z=14, x+2y−0z=5

Verified step by step guidance

1. Write down the system of equations clearly: 1) $x+0y+2z=11$, 2) $x+0y+3z=14$, 3) $x+2y-0z=5$.
2. Observe that the second term in the first two equations is zero, so the variable y does not appear in those equations. This means we can focus on x and z first.
3. Subtract the first equation from the second to eliminate x: $(x\; +\; 0y\; +\; 3z)\; -\; (x\; +\; 0y\; +\; 2z)\; =\; 14\; -\; 11$. Simplify this to find an equation involving only z.
4. Solve the resulting equation for z. Once you have z, substitute it back into either the first or second equation to solve for x.
5. Finally, substitute the values of x and z into the third equation $x\; +\; 2y\; =\; 5$ to solve for y.

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 a single point, infinitely many points, or no solution.

Methods for Solving Systems

Common methods to solve systems include substitution, elimination, and matrix techniques like Gaussian elimination. These methods help reduce the system to simpler forms to find variable values efficiently.

Interpreting Coefficients and Variables

Understanding the role of coefficients and variables in each equation is crucial. Zero coefficients indicate the absence of a variable in that equation, simplifying the system and guiding the choice of solution method.