Problem 27 - College Algebra with Modeling and Visualization

Question

If possible, solve the system. $$ \begin{array}{rr} -x+\quad 2 z= -9 \\ y+4 z= -13 \\ 3 x+y \quad =13 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 8\), \(y = -11\), and \(z = -\frac{1}{2}\).

Step 1: Analyze the System of Equations

We have three equations with three variables: \( x \), \( y \), and \( z \). The equations are given as: \(-x + 2z = -9\), \(y + 4z = -13\), and \(3x + y = 13\). The task is to solve for \(x\), \(y\), and \(z\).

Step 2: Express \(x\) in Terms of \(z\) from First Equation

From the first equation \(-x + 2z = -9\), we can express \(x\) in terms of \(z\): \(x = 2z + 9\).

Step 3: Express \(y\) in Terms of \(z\) from Second Equation

From the second equation \(y + 4z = -13\), we can express \(y\) in terms of \(z\): \(y = -4z - 13\).

Step 4: Substitute \(x\) and \(y\) into the Third Equation

Substitute \(x = 2z + 9\) and \(y = -4z - 13\) into the third equation \(3x + y = 13\). This gives: \(3(2z + 9) + (-4z - 13) = 13\). Simplify to find \(z\).

Step 5: Solve for \(z\)

Solving the equation \(3(2z + 9) + (-4z - 13) = 13\), we have: \(6z + 27 - 4z - 13 = 13\). Simplifying, we get \(2z + 14 = 13\). Solving for \(z\), we establish that \(z = -\frac{1}{2}\).

Step 6: Substitute \(z\) back to Find \(x\)

Substitute \(z = -\frac{1}{2}\) back into the expression for \(x\): \(x = 2(-\frac{1}{2}) + 9\), which simplifies to \(x = 8\).

Step 7: Substitute \(z\) back to Find \(y\)

Substitute \(z = -\frac{1}{2}\) back into the expression for \(y\): \(y = -4(-\frac{1}{2}) - 13\), which simplifies to \(y = -11\).

Step 8: Solution Verification

Verify the solution \((x, y, z) = (8, -11, -\frac{1}{2})\) by substituting back into the original equations. All equations hold true, confirming the solution.

Key Concepts

Solving Linear Equations

When solving linear equations, the goal is to find the values of the variables that hold true in the given equations. In the system we're looking at, we have three linear equations with three variables. These equations can be visualized as a set of straight lines, and we're searching for the point where all these lines intersect. Linear equations like

• \(-x + 2z = -9\)
• \(y + 4z = -13\)
• \(3x + y = 13\)

can be solved by finding common solutions for \(x\), \(y\), and \(z\). By systematically expressing one variable in terms of another and substituting back, we enable a step-by-step simplification, gradually revealing the values of our variables. Each substitution helps eliminate variables from the equations until reaching a solvable equation with just one variable.

Substitution Method

The substitution method is a powerful tool for solving systems of linear equations. Here's how it works in practice. The idea is to solve one of the equations for one variable, then substitute that expression into the other equations. For our given system:

1. We first express \(x\) in terms of \(z\) using the first equation: \(-x + 2z = -9\) becomes \(x = 2z + 9\).
2. Next, we express \(y\) in terms of \(z\) from the second equation: \(y + 4z = -13\) transforms to \(y = -4z - 13\).
3. Now, substitute these expressions for \(x\) and \(y\) into the third equation: \(3x + y = 13\).

This sequential substitution process simplifies the equations until we can solve for one variable, in this case, \(z\). Each step carefully reduces the number of unknown variables, leading to an easier resolution and eventually obtaining numerical values for all variables.

Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables, and operators. They form the foundation of setting up and solving equations. In our system of linear equations, the algebraic expressions represent relationships between the variables. For example, \(-x + 2z = -9\) is an algebraic expression showing a balance between the values substituted for \(x\) and \(z\).

Manipulating these expressions is key to solving the equations. When we substitute one variable's expression into another equation, we rely on the properties of equality and arithmetic operations:

• Addition and subtraction to isolate terms.
• Multiplication and division to rearrange and simplify expressions.

Ensuring each operation maintains the equality of both sides, we can reveal the unseen relationships between the variables, ultimately providing their numeric values. Understanding how to create, manipulate, and interpret algebraic expressions is a fundamental skill in algebra and crucial for problem-solving with equations.