6y 1.5 X 8 Solve For Y

Introduction to Linear Equations

Linear equations form the foundation of algebra and are essential tools in mathematics, science, engineering, and economics. They represent relationships between variables that change at constant rates. When we encounter an equation like 6y + 1.5x = 8 and are asked to solve for y, we're being asked to express y in terms of x, which is a fundamental algebraic skill with numerous practical applications.

Understanding the Equation

The equation 6y + 1.5x = 8 is a linear equation with two variables, y and x. Our goal is to isolate y on one side of the equation, expressing it in terms of x. This process will allow us to understand how y changes as x changes, which is particularly useful when graphing the equation or using it in applications.

Components of the Equation

In this equation:

• 6 is the coefficient of y
• 1.5 is the coefficient of x
• 8 is the constant term

Why Solve for y?

Solving for y allows us to express the equation in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. This form is particularly useful for graphing and understanding the relationship between variables.

Step-by-Step Solution

Let's solve the equation 6y + 1.5x = 8 for y systematically.

Step 1: Is

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Step 2: Subtract 1.5x from both sides

To isolate the term containing y, we need to move the 1.5x term to the other side of the equation. This is done by subtracting 1.5x from both sides:

6y + 1.5x - 1.5x = 8 - 1.5x

Simplifying this gives:

6y = -1.5x + 8

Step 3: Divide both sides by 6

Now, we have 6y equal to an expression involving x. To solve for y, we divide both sides of the equation by 6:

(6y)/6 = (-1.5x + 8)/6

Simplifying this yields:

y = (-1.5/6)x + 8/6

Step 4: Simplify the fractions

The coefficients can be simplified:

• -1.5 / 6 = -0.25 (or equivalently -1/4)
• 8 / 6 = 4/3 (simplified by dividing numerator and denominator by 2)

Therefore, the equation simplifies to:

y = -0.25x + 4/3

The Result and Its Significance

The solution y = -0.25x + 4/3 is the equation expressed in slope-intercept form (y = mx + b). Here:

• m = -0.25 is the slope of the line, indicating that for every unit increase in x, y decreases by 0.25.
• b = 4/3 is the y-intercept, the value of y when x = 0.

This form is highly valuable. It immediately reveals the line's steepness and direction (negative slope means it slopes downwards). It also allows for easy graphing: starting at the point (0, 4/3) and using the slope to find other points. Furthermore, this representation is fundamental for modeling relationships where one quantity changes linearly in response to another, such as cost versus quantity produced, distance versus time at constant speed, or temperature versus pressure in certain ranges.

Conclusion

Solving the linear equation 6y + 1.5x = 8 for y transforms it from a more complex form into the clear and practical slope-intercept form y = -0.25x + 4/3. This process, involving isolating the variable term and simplifying coefficients, is a core algebraic technique. The resulting equation provides immediate insight into the relationship between x and y, specifically the constant rate of change (slope) and the starting point (y-intercept). Mastering this skill is essential for interpreting

...linear relationships in diverse fields, from economics and physics to engineering and data science. By converting equations into this standardized format, complex problems become approachable, allowing for quick visual assessments and straightforward calculations of key characteristics like rate of change and initial value.

In essence, the ability to rearrange linear equations into slope-intercept form is more than an algebraic exercise; it is a fundamental literacy in quantitative reasoning. It empowers individuals to decode the story told by a line—whether that story describes a business's profit trend, a vehicle's motion, or a scientific principle—and to communicate that story with clarity and precision. Therefore, proficiency in this transformation is a cornerstone of mathematical competence and practical problem-solving.

Conclusion

Transforming the equation 6y + 1.5x = 8 into the slope-intercept form y = -0.25x + 4/3 distills its essence into two immediately interpretable parameters: a slope of -0.25 and a y-intercept of 4/3. This systematic process of algebraic manipulation—isolating the dependent variable and simplifying—reveals the linear relationship's core behavior. The resulting form serves as a powerful, universal key for graphing, prediction, and comparative analysis across countless real-world scenarios. Mastery of this technique is therefore indispensable for anyone seeking to model, understand, and articulate linear dependencies in both academic and practical contexts.