5x 2y 10 In Slope Intercept Form

5x + 2y = 10 in Slope-Intercept Form: A Comprehensive Guide

The equation 5x + 2y = 10 represents a linear relationship between two variables, x and y. While this is presented in standard form, converting it to slope-intercept form (y = mx + b) offers significant advantages in understanding its graphical representation and properties. This comprehensive guide will walk you through the process of converting 5x + 2y = 10 into slope-intercept form, exploring the meaning of the slope (m) and the y-intercept (b), and demonstrating how to graph the equation. We will also delve into practical applications and related concepts to ensure a thorough understanding.

Understanding Slope-Intercept Form (y = mx + b)

Before we begin the conversion, let's review the components of the slope-intercept form:

• y: Represents the dependent variable. Its value depends on the value of x.
• m: Represents the slope of the line. The slope describes the steepness and direction of the line. A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend. The slope is calculated as the change in y divided by the change in x (rise over run).
• x: Represents the independent variable. Its value can be chosen freely.
• b: Represents the y-intercept. This is the point where the line crosses the y-axis (where x = 0).

Converting 5x + 2y = 10 to Slope-Intercept Form

The goal is to isolate y on one side of the equation. Here's the step-by-step process:

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

2. Divide both sides by 2: This isolates y, giving us the slope-intercept form. 2y / 2 = (-5x + 10) / 2 This simplifies to: y = -\frac{5}{2}x + 5

Therefore, the equation 5x + 2y = 10 in slope-intercept form is y = -5/2x + 5.

Interpreting the Slope and y-intercept

Now that we have the equation in slope-intercept form, we can readily interpret its components:

• Slope (m = -5/2): The slope is -5/2. This means that for every 2 units increase in x, y decreases by 5 units. The negative sign indicates a downward trend from left to right. The slope signifies the rate of change of y with respect to x.

• y-intercept (b = 5): The y-intercept is 5. This means the line intersects the y-axis at the point (0, 5). This is the value of y when x is 0.

Graphing the Equation

Graphing the equation is straightforward using the slope and y-intercept:

1. Plot the y-intercept: Start by plotting the point (0, 5) on the y-axis.

2. Use the slope to find another point: The slope is -5/2. From the y-intercept (0, 5), move 2 units to the right (positive x direction) and 5 units down (negative y direction). This gives you the point (2, 0).

3. Draw the line: Draw a straight line through the two points (0, 5) and (2, 0). This line represents the equation 5x + 2y = 10.

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 original equation or the slope-intercept form:

Using the slope-intercept form:

0 = -5/2x + 5 5/2x = 5 x = 2

Therefore, the x-intercept is (2, 0).

Parallel and Perpendicular Lines

Understanding the slope allows us to determine relationships between lines.

• Parallel Lines: Parallel lines have the same slope. Any line parallel to y = -5/2x + 5 will also have a slope of -5/2.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -5/2 is 2/5. Any line perpendicular to y = -5/2x + 5 will have a slope of 2/5.

Real-World Applications

Linear equations like 5x + 2y = 10 have numerous real-world applications:

• Cost Analysis: Imagine x represents the number of units produced and y represents the total cost. The equation could model a scenario where the fixed cost is 5 (the y-intercept) and the cost per unit is 5/2.

• Distance-Time Relationships: x could represent time and y could represent distance. The equation might model the distance traveled by an object with a certain speed and initial position.

• Mixture Problems: x and y could represent the amounts of two different substances being mixed, with the equation representing a constraint on their total volume or concentration.

• Supply and Demand: In economics, linear equations often model supply and demand curves. The equation could represent a simplified model of the relationship between the price of a good (x) and the quantity demanded (y).

Solving Systems of Equations

The equation 5x + 2y = 10 can be part of a system of equations. Solving a system of equations means finding the values of x and y that satisfy both equations simultaneously. This can be done using various methods like substitution, elimination, or graphical methods. For example, if we have another equation, such as x + y = 3, we can solve for x and y by substitution or elimination.

Further Exploration: Transformations and Variations

Let's consider some variations and transformations of the equation:

• Changing the y-intercept: If we change the constant term in the equation (the 10), the y-intercept will change, resulting in a parallel line. For example, 5x + 2y = 20 would have a y-intercept of 10.

• Changing the slope: Altering the coefficients of x and y will change the slope of the line. For example, 10x + 4y = 20 is equivalent to y = -5/2x + 5, showing the same line. However, 5x + 4y = 10 would have a different slope.

• Non-linear transformations: Applying non-linear functions to the equation will result in a non-linear relationship.

Conclusion

Converting the equation 5x + 2y = 10 into slope-intercept form, y = -5/2x + 5, provides a clearer understanding of its properties. The slope and y-intercept offer valuable insights into the line's characteristics, including its steepness, direction, and the point where it intersects the y-axis. Graphing the equation, understanding parallel and perpendicular lines, and exploring real-world applications all contribute to a more comprehensive understanding of linear relationships. This foundation is crucial for further study in algebra, calculus, and various fields that rely on mathematical modeling. This guide has provided a detailed and comprehensive overview, covering the core concepts and extending to practical applications and further explorations. Mastering this concept builds a strong base for more advanced mathematical concepts.