Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} 2 x+y =-4 \\ -2 y+4 z =0 \\ 3 x -2 z=-11 \end{array}\right. $$

Short answer

The solution is \( x = -3 \), \( y = 2 \), \( z = 1 \)

Step-by-step solution

Solve for y from the first equation

Start with the first equation: \[ 2x + y = -4 \]Rearrange to get: \[ y = -4 - 2x \]

Solve for z from the second equation

Use the second equation: \[ -2y + 4z = 0 \]Substitute \( y = -4 - 2x \): \[ -2(-4 - 2x) + 4z = 0 \]Simplify: \[ 8 + 4x + 4z = 0 \]Rearrange to get: \[ z = -2 - x \]

Substitute y and z into the third equation

Take the third equation: \[ 3x - 2z = -11 \]Substitute \( z = -2 - x \): \[ 3x - 2(-2 - x) = -11 \]Simplify: \[ 3x + 4 + 2x = -11 \]Combine like terms: \[ 5x + 4 = -11 \]Solve for \( x \): \[ 5x = -15 \]\[ x = -3 \]

Substitute x back to find y and z

To find y, substitute \( x = -3 \) back into \( y = -4 - 2x \): \[ y = -4 - 2(-3) \]\[ y = -4 + 6 \]\[ y = 2 \]To find z, substitute \( x = -3 \) back into \( z = -2 - x \): \[ z = -2 - (-3) \]\[ z = -2 + 3 \]\[ z = 1 \]

Verify the solution

Substitute \( x = -3 \), \( y = 2 \), \( z = 1 \) back into the original equations to ensure they are satisfied:1. \[ 2(-3) + 2 = -4 \] checks out.2. \[ -2(2) + 4(1) = 0 \] checks out.3. \[ 3(-3) - 2(1) = -11 \] checks out.

Key concepts

solving systems

When we deal with 'systems of equations,' we are looking at multiple equations that share common variables. Our goal is to find the correct values for these variables that satisfy all equations simultaneously. This type of problem appears in various areas like economics, engineering, and science.
For instance, the given problem is a system of three linear equations with three variables \( x \), \( y \), and \( z \). Solving a system starts by using techniques like substitution, elimination, or graphical methods. Let's dive deeper into the substitution method which we used here.

substitution method

The 'substitution method' is one of the simplest techniques to solve systems of equations. Here's a quick overview:

• Solve one of the equations for one variable.
• Substitute this expression into the other equations.
• Solve the new equation(s) to find the value of the second variable.
• Substitute back to find the remaining variable(s).

In our problem, we started by solving the first equation for \( y \):
\[ 2x + y = -4 \ \to y = -4 - 2x \]
We then substituted this expression into the second and third equations. Performing substitution can sometimes get a bit tricky, but practice makes perfect!

linear equations

Linear equations are equations of the first order, meaning their graph is a straight line. They generally look like this: \[ ax + by + cz + ... = d \] where \( a, b, \) and \( c \) are coefficients, and \( d \) is a constant. In our exercise, each equation was linear. For example:
\[ 3x - 2z = -11 \]
Solving linear equations may involve isolating variables, combining like terms, and performing arithmetic operations. These skills are essential for solving systems of linear equations efficiently.

step-by-step solution

Breaking down complex problems into a 'step-by-step solution' is crucial for understanding. Let's recap our approach for the given system:

• Step 1: Solve for \( y \) from the first equation: \[ y = -4 - 2x \]
• Step 2: Substitute \( y \) into the second equation and solve for \( z \): \[ z = -2 - x \]
• Step 3: Substitute \( y \) and \( z \) into the third equation to solve for \( x \): \[ x = -3 \]
• Step 4: Use the values of \( x \) to find \( y \) and \( z \): \[ y = 2, z = 1 \]
• Step 5: Verify the solution by substituting \( x, y, \) and \( z \) back into the original equations - they should all hold true.

Using a step-by-step methodical approach ensures that each variable is correctly calculated and validated. It eliminates errors and builds confidence as you work through the problem.