2x 5y 10 Solve For Y

Solving for 'y': A Deep Dive into 2x + 5y = 10

This article provides a comprehensive guide on how to solve the equation 2x + 5y = 10 for 'y'. We'll explore various methods, delve into the underlying algebraic principles, and examine practical applications. Understanding this seemingly simple equation unlocks a world of mathematical concepts, crucial for various fields like algebra, calculus, and even programming.

Understanding the Equation: 2x + 5y = 10

The equation 2x + 5y = 10 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 specific equation describes a relationship where the combination of 'x' and 'y' always results in a sum of 10, after applying the specified coefficients (2 and 5).

Key Terms and Concepts

Before we proceed, let's clarify some essential terminology:

• Variable: A symbol (usually a letter like x or y) representing an unknown quantity.
• Coefficient: The numerical factor multiplying a variable (e.g., 2 in 2x).
• Constant: A numerical value that stands alone in an equation (e.g., 10 in 2x + 5y = 10).
• Linear Equation: An equation where the highest power of any variable is 1.
• Solving for a Variable: Isolating a specific variable on one side of the equation to express it in terms of other variables or constants.

Methods to Solve for 'y'

There are several ways to solve 2x + 5y = 10 for 'y'. We'll explore two common and effective approaches:

Method 1: Algebraic Manipulation

This method involves systematically manipulating the equation using basic algebraic principles to isolate 'y'. The steps are as follows:

1. Subtract 2x from both sides: This moves the term containing 'x' to the right side of the equation. 2x + 5y - 2x = 10 - 2x This simplifies to: 5y = 10 - 2x

2. Divide both sides by 5: This isolates 'y' by eliminating its coefficient. 5y / 5 = (10 - 2x) / 5 This simplifies to: y = 2 - (2/5)x or y = - (2/5)x + 2

This final equation, y = -(2/5)x + 2, expresses 'y' explicitly in terms of 'x'. This is the slope-intercept form of a linear equation (y = mx + c), where 'm' represents the slope (-2/5 in this case) and 'c' represents the y-intercept (2).

Method 2: Using the Properties of Equality

This method emphasizes the properties of equality to maintain balance while solving for 'y'.

1. Subtraction Property of Equality: Subtract 2x from both sides. This property states that subtracting the same value from both sides of an equation maintains equality. 2x + 5y - 2x = 10 - 2x 5y = 10 - 2x

2. Division Property of Equality: Divide both sides by 5. This property states that dividing both sides of an equation by the same non-zero value maintains equality. 5y / 5 = (10 - 2x) / 5 y = 2 - (2/5)x

This method highlights the underlying principles governing algebraic manipulation, reinforcing the understanding of why the steps work.

Interpreting the Solution: y = -(2/5)x + 2

The solution, y = -(2/5)x + 2, reveals several crucial pieces of information:

• Slope: The slope, -2/5, indicates the rate of change of 'y' with respect to 'x'. For every 5-unit increase in 'x', 'y' decreases by 2 units. The negative slope signifies a downward trend in the line's graph.

• Y-intercept: The y-intercept, 2, represents the point where the line intersects the y-axis (when x = 0). This means when x is 0, y is 2.

• Graphical Representation: The equation represents a straight line. Plotting points satisfying this equation will reveal a straight line with a slope of -2/5 and a y-intercept of 2.

Applications and Further Exploration

The ability to solve for 'y' in equations like 2x + 5y = 10 is fundamental to many areas:

1. Graphing Linear Equations:

Solving for 'y' allows easy graphing. The slope-intercept form (y = mx + c) directly provides the slope and y-intercept, enabling quick plotting.

2. System of Equations:

When dealing with a system of linear equations (multiple equations with multiple variables), solving for one variable in terms of others facilitates substitution methods, leading to solutions for the entire system.

3. Data Analysis and Modeling:

Linear equations are often used to model relationships between variables in real-world scenarios. Solving for a specific variable allows for predictions and analysis based on different input values. For example, if 'x' represents the number of hours worked and 'y' represents earnings, solving for 'y' gives the earnings based on the number of hours worked.

4. Programming and Computer Science:

Linear equations and their solutions form the basis of many algorithms and computations in computer science. Manipulating equations programmatically often involves solving for specific variables.

5. Calculus and Beyond:

Understanding linear equations and solving for variables builds a strong foundation for more advanced mathematical concepts like calculus, differential equations, and linear algebra.

Expanding on the Concepts

Let's explore some related concepts to solidify your understanding:

Finding x-intercept

The x-intercept is the point where the line crosses the x-axis (where y = 0). To find it, substitute y = 0 into the equation:

2x + 5(0) = 10 2x = 10 x = 5

The x-intercept is (5, 0).

Solving for x

We can also solve the original equation for x:

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

This expresses x in terms of y.

Conclusion

Solving for 'y' in the equation 2x + 5y = 10 is a fundamental algebraic skill. By mastering this, you've unlocked a gateway to understanding linear equations, their graphical representations, and their practical applications across various disciplines. Remember the steps, understand the underlying principles, and practice regularly to build confidence and proficiency. The more you explore these concepts, the more you'll appreciate their power and versatility in problem-solving. This seemingly simple equation serves as a solid foundation for more complex mathematical endeavors.