Problem 47 - Elementary Algebra

Question

Solve \(x-3 y+5 z+1=2 y-7 z+8\) for \(x\).

Step-by-Step Solution

Answer

Question: Given the equation \(x-3y+5z+1=2y-7z+8\), solve for x. Answer: \(x=5y-12z+7\)

Step 1: Write down the given equation

The given equation is \(x-3y+5z+1=2y-7z+8\).

Step 2: Move "y" terms to the right side of the equation

To do this, add \(3y\) to both the sides of the equation: \(x+5z+1+3y=2y-7z+8+3y\).

Step 3: Simplify the equation

This step gives: \(x+5z+1+3y=5y-7z+8\).

Step 4: Move the "z" terms to the right side of the equation

Subtract \(5z\) from both sides: \(x+1=5y-7z+8-5z\).

Step 5: Move the constant term to the right side of the equation

Subtract \(1\) from both sides of the equation: \(x=5y-12z+7\).

Step 6: Write down the final equation for x

The solution is \(x=5y-12z+7\).

Key Concepts

Understanding Linear Equations

Linear equations are foundational concepts in algebra. They are equations where each term is either a constant or the product of a constant and a single variable. A linear equation forms a straight line when graphed on a coordinate plane. In this exercise, we work with a 3-variable linear equation in the form of:

• \(x - 3y + 5z + 1 = 2y - 7z + 8\)

The goal is to manipulate this equation and isolate one variable—\(x\)—on one side, highlighting the linear relationship among variables \(x, y,\) and \(z\).

Mastering Variables Manipulation

Variables manipulation involves rearranging and simplifying equations to isolate a variable of interest. This exercise requires moving terms intelligently across the equation.
To balance both sides and solve for \(x\), we must understand operations that maintain equality:

• Adding or subtracting the same value from both sides.
• Moving terms, like \(y\) and \(z\), responsibly by performing inverse operations.

By adding \(3y\) to both sides and rearranging terms, we skillfully shift the \(y\) terms to one side, paving the way to a clearer manipulation.

Effective Equation Solving Techniques

Equation solving is the heart of algebra. In order to isolate \(x\), we consistently apply balancing and inverse operations:

• Balancing Terms: Add or subtract to cancel out unwanted variables from one side.
• Re-arranging Steps: Simplify step-by-step, focusing on one variable movement at a time.

In this problem, after organizing the \(y\) terms, we move to tackle the \(z\) components by subtracting \(5z\) from both sides. This effective technique will eventually lead us to a neat solution for \(x\).

Step-by-Step Solution to Solve the Equation

Following a step-by-step approach ensures accuracy and understanding. Here, we've outlined the primary steps utilized:

1. Start with The Original Equation: Write down the given problem initially.
2. Move \(y\) terms: Shift \(y\) terms methodically to one side.
3. Simplify: Once rearranged, simplify to clear up cluttered terms.
4. Focus on \(z\): Switch \(z\) terms next, employing subtraction if necessary.
5. Conclusion: Isolate \(x\) for the final simplified equation, \(x=5y-12z+7\).

This logical progression helps framers understand transitioning algebraic equations into simpler forms, making it possible to solve for specific variables.