Y 4x Rx 6 Solve For X

Solving for x: A Comprehensive Guide to y = 4x + rx - 6

This article provides a detailed explanation of how to solve the equation y = 4x + rx - 6 for x. We'll explore different approaches, discuss the importance of understanding the underlying mathematical concepts, and offer practical examples to solidify your understanding. This guide caters to various levels of mathematical proficiency, from beginners seeking a fundamental understanding to those aiming to refine their algebraic skills.

Understanding the Equation: y = 4x + rx - 6

Before diving into the solution, let's break down the equation itself: y = 4x + rx - 6. This is a linear equation, meaning the highest power of x is 1. It contains three key elements:

• y: This represents the dependent variable, meaning its value depends on the value of x.
• x: This is the independent variable. We aim to solve the equation for x, meaning we want to express x in terms of y and r.
• 4, r, and -6: These are constants. '4' and 'r' are coefficients of x, while '-6' is a constant term. Note that 'r' is a parameter; its value can change, influencing the solution.

The presence of 'r' makes this equation slightly more complex than a simple linear equation where all coefficients are numerical values. However, the solution strategy remains fundamentally the same.

Method 1: Factoring and Solving

This method involves factoring out x from the terms containing it, and then isolating x.

Step 1: Factor out x:

We observe that both 4x and rx contain the variable x. Therefore, we can factor x out:

y = x(4 + r) - 6

Step 2: Isolate the term with x:

To isolate the term containing x, we add 6 to both sides of the equation:

y + 6 = x(4 + r)

Step 3: Solve for x:

Finally, we divide both sides by (4 + r) to solve for x:

x = (y + 6) / (4 + r)

Important Consideration: This solution is valid only if (4 + r) ≠ 0. If (4 + r) = 0, then the equation simplifies to y = -6, and x can take on any value. This represents a vertical line, meaning there are infinitely many solutions. This condition (4 + r ≠ 0) is crucial and demonstrates a key concept in algebra: division by zero is undefined.

Method 2: Rearranging and Solving

This method involves rearranging the equation step-by-step to isolate x.

Step 1: Combine like terms:

The equation is already partially simplified, but let's combine the terms containing x:

y = (4 + r)x - 6

Step 2: Add 6 to both sides:

Adding 6 to both sides will move the constant term to the left-hand side:

y + 6 = (4 + r)x

Step 3: Divide by (4 + r):

Dividing both sides by (4 + r) isolates x, leading to the same solution as Method 1:

x = (y + 6) / (4 + r)

Again, the condition (4 + r) ≠ 0 must hold for this solution to be valid.

Understanding the Solution: x = (y + 6) / (4 + r)

The solution x = (y + 6) / (4 + r) tells us that the value of x depends on the values of y and r. It's a formula that allows us to calculate x given any values for y and r (provided that r ≠ -4).

• Impact of y: As y increases, x will also increase (assuming 4 + r is positive). If 4 + r is negative, then as y increases, x will decrease.
• Impact of r: The value of r directly influences the slope of the line represented by the equation. Changes in r will affect the rate at which x changes in response to changes in y.

Practical Examples

Let's illustrate the solution with some numerical examples.

Example 1: y = 10, r = 2

Substitute y = 10 and r = 2 into the solution:

x = (10 + 6) / (4 + 2) = 16 / 6 = 8/3

Therefore, when y = 10 and r = 2, x = 8/3.

Example 2: y = -2, r = -1

Substitute y = -2 and r = -1 into the solution:

x = (-2 + 6) / (4 + (-1)) = 4 / 3

Therefore, when y = -2 and r = -1, x = 4/3.

Example 3: Illustrating the Undefined Case (r = -4)

Let's see what happens when r = -4:

x = (y + 6) / (4 + (-4)) = (y + 6) / 0

Division by zero is undefined. This confirms our earlier observation that the solution is only valid when r ≠ -4. In this case, the equation becomes:

y = 4x - 4x - 6 y = -6

This represents a horizontal line where y is always -6, regardless of the value of x. Therefore, there are infinitely many solutions for x.

Advanced Considerations: Graphical Representation

The equation y = 4x + rx - 6 represents a straight line on a Cartesian coordinate system. The slope of this line is (4 + r), and the y-intercept is -6. Understanding the graphical representation provides a visual interpretation of the solution.

• Different values of r will result in lines with different slopes, all passing through the same y-intercept (-6).
• The case where r = -4 results in a horizontal line at y = -6, visually demonstrating the infinite solutions for x.

Using graphing software or by hand-plotting points, you can visualize how the line changes as the value of 'r' varies, providing a deeper understanding of the relationship between the variables.

Conclusion: Mastering the Solution

Solving the equation y = 4x + rx - 6 for x requires a solid understanding of basic algebraic manipulation, specifically factoring and solving linear equations. The solution, x = (y + 6) / (4 + r), provides a formula to calculate x given values for y and r, provided r ≠ -4. Remembering the condition that (4 + r) ≠ 0 is crucial for the validity of the solution. This problem highlights the importance of considering potential undefined cases in algebraic solutions. By practicing these methods and exploring the graphical representation, you can build confidence and proficiency in solving similar algebraic equations. Remember to always double-check your work and consider the limitations of your solutions.