Problem 11 - College Algebra

Question

For the following exercises, solve each system by substitution. $$ \begin{aligned} 3 x-4 y+2 z &=-15 \\ 2 x+4 y+z &=16 \\ 2 x+3 y+5 z &=20 \end{aligned} $$

Step-by-Step Solution

Answer

\(x = -1, y = 4, z = 2\)

Step 1: Choose an Equation for Substitution

Begin with the second equation, \(2x + 4y + z = 16\), which is the simplest to manipulate for substitution. We'll solve for \(z\) in terms of \(x\) and \(y\).

Step 2: Solve for z

From the equation \(2x + 4y + z = 16\), isolate \(z\) to get:\[ z = 16 - 2x - 4y \]

Step 3: Substitute z into Other Equations

Replace \(z\) in the first and third equations with the expression \(16 - 2x - 4y\) from Step 2.1st equation: \[ 3x - 4y + 2(16 - 2x - 4y) = -15 \] 3rd equation:\[ 2x + 3y + 5(16 - 2x - 4y) = 20 \]

Step 4: Simplify the First Equation

Simplify the expression from the first equation:\[ 3x - 4y + 32 - 4x - 8y = -15 \] Combine like terms:\[ -x - 12y = -47 \] This simplifies further to:\[ x + 12y = 47 \]

Step 5: Simplify the Third Equation

Simplify the expression from the third equation:\[ 2x + 3y + 80 - 10x - 20y = 20 \] Combine like terms:\[ -8x - 17y = -60 \] This simplifies further to:\[ 8x + 17y = 60 \]

Step 6: Solve the New System of Equations

Now, solve the two linear equations:1. \(x + 12y = 47\)2. \(8x + 17y = 60\)From equation 1, solve for \(x\):\[ x = 47 - 12y \] Substitute this into equation 2:\[ 8(47 - 12y) + 17y = 60 \]

Step 7: Simplify and Solve for y

Expand and simplify the substituted equation:\[ 376 - 96y + 17y = 60 \] Combine like terms:\[ -79y = -316 \] Solving for \(y\):\[ y = 4 \]

Step 8: Solve for x using y

Use \(y = 4\) in the solved equation from Step 6:\[ x = 47 - 12(4) \] This simplifies to:\[ x = 47 - 48 \]\[ x = -1 \]

Step 9: Solve for z using x and y

Substitute \(x = -1\) and \(y = 4\) into the expression for \(z\) from Step 2:\[ z = 16 - 2(-1) - 4(4) \] This simplifies to:\[ z = 16 + 2 - 16 \]\[ z = 2 \]

Step 10: Conclusion: Solution to the System

The solution to the system of equations is \(x = -1, y = 4, z = 2\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a core technique used in solving systems of equations by substitution. This method involves rearranging equations to isolate variables and substitute them into other equations. When you start with an equation like \(2x + 4y + z = 16\), you need to rearrange it to solve for one variable. This helps to reduce complexity and facilitates substitution into other equations.
For instance, isolating \(z\) gives us \(z = 16 - 2x - 4y\). By substituting this expression for \(z\) in the other equations, we can simplify them and eliminate one variable at a time.

• Isolation: Find the variable to isolate and use basic operations to express it in terms of other variables.
• Substitution: Replace the isolated variable in the other equations to simplify them.
• Combination: Use algebraic operations to combine like terms and reduce equations.

Mastering algebraic manipulation is crucial as it allows you to systematically break down and solve even complicated systems of equations.

Linear Equations

Linear equations form the backbone of mathematical problem-solving methods like substitution. These are equations of the first degree, which means they don't involve variables raised to a power higher than one. A system of linear equations is a collection of two or more linear equations that have a common solution.
In our original exercise, we dealt with three equations:

• \(3x - 4y + 2z = -15\)
• \(2x + 4y + z = 16\)
• \(2x + 3y + 5z = 20\)

The goal is to find values for \(x\), \(y\), and \(z\) that satisfy all these equations simultaneously. This often involves:

• Reordering equations to make one variable easier to eliminate across the equations.
• Using substitution to express dependencies between variables.
• Simplifying the resulting equations to linear simple forms that reveal the solutions.

Understanding the structure of linear equations helps you predict the number of solutions possible and the best strategy to adopt.

Mathematical Problem Solving

Mathematical problem-solving through substitution and solving systems of equations is a strategic process. It requires selecting an appropriate equation to manipulate, systematically applying algebraic principles, and checking your solutions for consistency with the original equations.
To effectively solve a system of linear equations, follow these strategies:

• Identify the simplest equation for substitution. This reduces the chance of making errors and simplifies the calculations.
• Perform substitutions carefully to avoid introducing errors. Use careful notation and verify each step as you proceed.
• Once simplified, solve the resulting equations step by step, validating your solutions against the initial equations to ensure they are correct.

When you arrive at a solution like \((x = -1, y = 4, z = 2)\), always ensure your final step is to substitute the values back into the original equations to confirm your answers satisfy all conditions.