6 4x 9y Solve For Y

Solving for 'y': A Comprehensive Guide to 6 = 4x + 9y

This article provides a detailed, step-by-step guide on how to solve the equation 6 = 4x + 9y for 'y'. We'll explore various approaches, discuss the significance of the solution, and delve into related concepts to provide a thorough understanding of the topic. This will cover algebraic manipulation, the concept of linear equations, and the implications of different solution methods.

Understanding the Equation: 6 = 4x + 9y

The equation 6 = 4x + 9y represents a linear equation in two variables, 'x' and 'y'. A linear equation is characterized by its straight-line graph when plotted on a Cartesian coordinate system. This particular equation describes a line with a specific slope and y-intercept. Solving for 'y' means isolating 'y' on one side of the equation, expressing it in terms of 'x'. This gives us a formula to calculate the value of 'y' for any given value of 'x'.

Why Solve for 'y'?

Solving for 'y' is crucial for several reasons:

Finding y-intercept: When x = 0, the equation directly gives the y-intercept, the point where the line intersects the y-axis.
Understanding the relationship between x and y: The solved equation shows how 'y' changes with changes in 'x'. This reveals the direct relationship between the two variables.
Graphing the equation: An equation solved for 'y' (in the form y = mx + c) makes it much easier to graph the line. The coefficient of x (m) is the slope, and the constant term (c) is the y-intercept.
Further calculations and analysis: Having 'y' isolated simplifies any subsequent calculations or analyses involving the equation.

Solving for 'y': Step-by-Step Guide

Here's a detailed walkthrough of how to solve 6 = 4x + 9y for 'y':

Step 1: Isolate the term containing 'y'.

To isolate the term with 'y', we need to move the term with 'x' to the other side of the equation. We do this by subtracting 4x from both sides:

6 - 4x = 9y

Step 2: Solve for 'y' by dividing.

Now, the term with 'y' is isolated. To solve for 'y', we divide both sides of the equation by 9:

(6 - 4x) / 9 = y

Step 3: Simplify (Optional).

While the equation is solved, we can simplify the expression for better readability. We can separate the fraction:

(6/9) - (4x/9) = y

Further simplification yields:

(2/3) - (4/9)x = y

Therefore, the solution for 'y' is:

y = (2/3) - (4/9)x or y = - (4/9)x + (2/3)

Alternative Approach: Using the Standard Form

The given equation, 6 = 4x + 9y, is in a slightly unconventional form. Linear equations are often presented in the standard form: Ax + By = C, where A, B, and C are constants. Let's rearrange the given equation into this standard form:

4x + 9y = 6

Following the same steps as before:

Subtract 4x from both sides: 9y = -4x + 6
Divide both sides by 9: y = (-4/9)x + (6/9)
Simplify: y = (-4/9)x + (2/3)

This demonstrates that regardless of the initial presentation, the solution remains the same.

Graphical Representation and Interpretation

The equation y = (-4/9)x + (2/3) represents a straight line. Let's analyze its key features:

Slope: The slope, -4/9, indicates the steepness and direction of the line. A negative slope means the line descends from left to right. The slope tells us that for every 9-unit increase in x, y decreases by 4 units.
y-intercept: The y-intercept is 2/3. This is the point where the line crosses the y-axis (when x = 0).

By plotting these points and drawing a line through them, we can visualize the relationship between x and y defined by the equation.

Applications and Extensions

The ability to solve for 'y' in linear equations has numerous practical applications across various fields:

Economics: Supply and demand curves are often represented by linear equations. Solving for 'y' helps determine the quantity demanded or supplied at a given price.
Physics: Equations describing motion, such as velocity and displacement, are frequently linear. Solving for 'y' (or a similar variable) provides crucial information about the physical system.
Engineering: Linear equations are extensively used in structural analysis and circuit design. Solving for variables like current or voltage often involves isolating specific variables.
Computer graphics: Linear equations form the foundation of many graphical transformations and projections used in computer graphics and animation.

Further Exploration: Systems of Equations

Understanding how to solve for 'y' in a single linear equation forms a fundamental building block for solving systems of linear equations. A system of linear equations involves two or more equations with the same variables. Solving these systems requires techniques like substitution or elimination to find the values of the variables that satisfy all equations simultaneously. The solution for 'y' from a single equation can often be substituted into another equation to solve for the remaining variable(s).

Conclusion: Mastering the Fundamentals

Solving the equation 6 = 4x + 9y for 'y' is more than just an algebraic exercise. It reinforces the understanding of linear equations, their graphical representations, and their applications in various fields. Mastering this fundamental skill is crucial for tackling more complex mathematical problems and for successfully applying mathematical principles to real-world situations. The detailed, step-by-step approach provided in this article aims to build a solid foundation for anyone seeking to improve their algebraic skills and broaden their mathematical understanding. Remember the key steps: isolate the term with 'y', then divide to solve for 'y', and simplify the result where possible. This straightforward process unlocks a deeper understanding of linear relationships and provides a springboard for tackling more advanced mathematical concepts.