Solve For Y 2x 5y 10

Solve for y: 2x + 5y = 10 – A Comprehensive Guide

This seemingly simple equation, 2x + 5y = 10, opens the door to a world of algebraic understanding. Solving for 'y' isn't just about finding a numerical answer; it's about mastering fundamental algebraic manipulation techniques applicable to countless mathematical problems. This article will explore multiple approaches to solving for 'y', delve into the significance of the solution, and illustrate its application in real-world scenarios.

Understanding the Equation: 2x + 5y = 10

Before jumping into the solution, let's break down the equation itself. This is a linear equation in two variables, 'x' and 'y'. A linear equation represents a straight line when graphed. The equation is written in standard form, Ax + By = C, where A, B, and C are constants. In our case, A = 2, B = 5, and C = 10.

Method 1: Isolating 'y' Through Algebraic Manipulation

The most direct method for solving for 'y' involves isolating the 'y' term on one side of the equation. This is achieved through a series of algebraic operations, maintaining the equation's balance:

1. Subtract 2x from both sides: This eliminates the 'x' term from the left side, leaving only terms involving 'y'.

   2x + 5y - 2x = 10 - 2x
   5y = 10 - 2x
   

2. Divide both sides by 5: This isolates 'y' completely.

   5y / 5 = (10 - 2x) / 5
   y = (10 - 2x) / 5
   

3. Simplify (optional): The solution can be further simplified by distributing the division:

   y = 10/5 - (2x)/5
   y = 2 - (2/5)x
   

   Or, equivalently:

   y = 2 - 0.4x
   

This final expression, y = 2 - (2/5)x or y = 2 - 0.4x, shows 'y' explicitly as a function of 'x'. For any given value of 'x', we can calculate the corresponding value of 'y'.

Method 2: Using the Slope-Intercept Form

The equation can also be solved by transforming it into the slope-intercept form, y = mx + b, where 'm' is the slope and 'b' is the y-intercept. This form offers valuable insights into the line's graphical representation.

1. Start with the original equation: 2x + 5y = 10

2. Subtract 2x from both sides: 5y = -2x + 10

3. Divide both sides by 5: y = (-2/5)x + 2

Now, the equation is in slope-intercept form:

• Slope (m) = -2/5: This indicates the line's steepness and direction. A negative slope means the line slopes downward from left to right.

• Y-intercept (b) = 2: This is the point where the line intersects the y-axis (where x = 0).

This method not only solves for 'y' but also provides a clear geometrical interpretation of the equation.

Understanding the Solution: What Does it Mean?

The solution, y = 2 - (2/5)x, represents a relationship between 'x' and 'y'. It defines all the points (x, y) that lie on the straight line represented by the equation 2x + 5y = 10. This means that for every value of 'x', there's a corresponding value of 'y' that satisfies the equation.

For example:

• If x = 0, y = 2 - (2/5)(0) = 2
• If x = 5, y = 2 - (2/5)(5) = 0
• If x = -5, y = 2 - (2/5)(-5) = 4

Application in Real-World Scenarios

Linear equations like 2x + 5y = 10 are surprisingly common in various real-world situations. Consider these examples:

• Mixing solutions: Imagine you're mixing two liquids, where 'x' represents the volume of liquid A and 'y' represents the volume of liquid B. The equation might represent a constraint on the total volume or the concentration of a specific component in the mixture.

• Budgeting: 'x' could represent the cost of one item, and 'y' the cost of another. The equation might represent a budget constraint, where the total spending (2x + 5y) must equal a certain amount (10).

• Production planning: 'x' could represent the number of units of product A produced, and 'y' the number of units of product B. The equation could represent a resource constraint, such as the total available labor hours.

• Distance-Time problems: In simpler versions, 'x' and 'y' might represent the speeds of two objects, and the equation could be a relationship between their travel times and distances.

Graphical Representation

Graphing the equation helps visualize the solution. The line representing 2x + 5y = 10 passes through the points (0, 2) and (5, 0), as calculated earlier. Plotting these points and drawing a line through them visually demonstrates the infinite number of (x,y) pairs satisfying the equation. Online graphing tools or even simple graph paper can be used for this.

Advanced Concepts and Extensions

While solving for 'y' in this specific equation is relatively straightforward, the techniques employed extend to more complex scenarios:

• Systems of Equations: The ability to solve for a single variable is crucial when dealing with systems of linear equations (multiple equations with multiple variables). Techniques like substitution or elimination build upon the foundational steps shown above.

• Non-linear Equations: The principles of algebraic manipulation remain applicable even when dealing with equations that aren't linear (e.g., quadratic equations). The process of isolating a variable might be more involved but follows similar logic.

• Calculus: The concept of solving for one variable in terms of another is fundamental in calculus, particularly when dealing with implicit differentiation and related rates.

Conclusion

Solving for 'y' in the equation 2x + 5y = 10 is more than a simple algebraic exercise; it's a gateway to understanding fundamental mathematical principles applicable across various disciplines. By mastering this basic skill, one gains a deeper appreciation for linear equations, their graphical representation, and their wide-ranging applications in the real world. The techniques discussed—isolating variables, converting to slope-intercept form, and understanding the graphical interpretation—form a strong foundation for tackling more complex mathematical problems. Remember that practice is key. The more you work with these types of equations, the more comfortable and proficient you will become.