Answer:

Understanding the Equation y = -2x + 6

Introduction: The equation y = -2x + 6 represents a linear relationship between the variables x and y. In this answer, I will provide a clear explanation of this equation and its significance. Whether you’re a beginner or have little knowledge of this topic, I’ll break it down into simple terms for easy understanding.

Explanation: Here’s what you need to know about the equation y = -2x + 6:

- The equation is in slope-intercept form: y = mx + b. In this form, ‘m’ represents the slope of the line, and ‘b’ represents the y-intercept.
- In our equation, the slope ‘m’ is -2, indicating that for every increase of 1 unit in x, y decreases by 2 units.
- The y-intercept ‘b’ is 6, which means the line crosses the y-axis at the point (0, 6).

- Graphical representation:
- To visualize the equation, you can plot its points on a coordinate plane.
- Start by plotting the y-intercept (0, 6). This point indicates that when x is 0, y equals 6.
- From the y-intercept, use the slope to find additional points. For example:
- When x = 1, y = -2(1) + 6 = 4. Plot the point (1, 4).
- When x = -1, y = -2(-1) + 6 = 8. Plot the point (-1, 8).

- Connect these points with a straight line. The resulting line represents the equation y = -2x + 6.

- Meaning of the equation:
- The equation represents a linear relationship where y decreases by 2 units for every 1 unit increase in x.
- The negative slope indicates a downward trend, meaning as x increases, y decreases.
- The y-intercept at (0, 6) shows that when x is 0, y is 6. This point represents the initial value of y when x is not present.

Example applications: Understanding this equation can be helpful in various scenarios. Here are a few examples:

- Cost and revenue: Suppose you run a business where the cost of production increases by $2 for every additional unit produced (x). The equation y = -2x + 6 can represent your revenue (y) as a function of the number of units produced (x).
- Temperature and time: Imagine you have a cooling process where the temperature (y) decreases by 2 degrees Celsius for every hour (x) that passes. The equation y = -2x + 6 can describe the relationship between time and temperature during this process.
- Sports and performance: If you’re analyzing an athlete’s performance, the equation can be used to measure improvement. For instance, y could represent the number of goals scored in a game, while x represents the training time. The equation y = -2x + 6 would then depict the expected number of goals based on the time spent training.

Conclusion: In summary, the equation y = -2x + 6 represents a linear relationship with a negative slope. By understanding its slope-intercept form, graphical representation, and real-world applications, you can gain insights into how the variables x and y are related. Remember, this equation is just one example, and there are countless other equations that describe different relationships between variables.