Identifying Points On A Line Y + 8 = 4(x - 5)
In the realm of mathematics, understanding linear equations is a fundamental skill. Linear equations, with their straight-line representations, play a crucial role in various fields, from physics and engineering to economics and computer science. One common task when working with linear equations is determining whether a given point lies on the line described by the equation. This article delves into this concept, using the equation y + 8 = 4(x - 5) as an example. We will explore the equation, its properties, and how to verify which of the provided points – A. (4, 5), B. (-5, -8), C. (5, 0), D. (4, -8), E. (5, -8), and F. (5, 8) – satisfy the equation and thus lie on the line. Mastering this skill is essential for anyone seeking to build a strong foundation in algebra and beyond.
Understanding Linear Equations
Before we dive into the specifics of our equation, let's first establish a clear understanding of what linear equations are and why they matter. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variables in a linear equation can only have a power of one; no exponents, square roots, or other functions are allowed. When graphed on a coordinate plane, linear equations produce a straight line. This is where the term "linear" comes from.
Linear equations are typically written in one of several standard forms, each offering unique insights into the line's characteristics. The most common forms include:
- Slope-intercept form: y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis).
- Point-slope form: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a specific point on the line. This form is particularly useful when you know the slope and one point on the line.
- Standard form: Ax + By = C, where A, B, and C are constants. This form is often used for representing linear equations in a general way.
The slope of a line is a crucial concept. It measures the steepness and direction of the line. A positive slope indicates that the line is increasing (going upwards) as you move from left to right, while a negative slope indicates that the line is decreasing (going downwards). The slope is calculated as the change in y divided by the change in x between any two points on the line (rise over run).
Linear equations are not just abstract mathematical concepts; they have countless real-world applications. They can model relationships between quantities that change at a constant rate, such as the distance traveled by a car moving at a constant speed, the cost of renting a car based on mileage, or the relationship between temperature in Celsius and Fahrenheit. Understanding linear equations allows us to analyze and predict these types of relationships, making them invaluable tools in various disciplines.
Analyzing the Given Equation: y + 8 = 4(x - 5)
Now, let's turn our attention to the specific linear equation we are working with: y + 8 = 4(x - 5). This equation is presented in a form that closely resembles the point-slope form, which is a valuable clue for our analysis. To fully understand the equation, we will first manipulate it into slope-intercept form, which will reveal the slope and y-intercept of the line. This will give us a clearer picture of the line's behavior and characteristics.
To convert the equation to slope-intercept form (y = mx + b), we need to isolate y on one side of the equation. We can do this by following these steps:
- Distribute the 4 on the right side of the equation: y + 8 = 4x - 20
- Subtract 8 from both sides of the equation: y = 4x - 20 - 8
- Simplify: y = 4x - 28
Now, the equation is in slope-intercept form. We can easily identify the slope (m) as 4 and the y-intercept (b) as -28. This tells us that the line has a positive slope, meaning it increases as we move from left to right, and it crosses the y-axis at the point (0, -28).
Alternatively, we can recognize that the original equation, y + 8 = 4(x - 5), is in point-slope form, y - y1 = m(x - x1). By comparing the given equation with the point-slope form, we can directly identify the slope m as 4 and a point (x1, y1) on the line as (5, -8). This provides us with a quick way to determine a point that lies on the line without converting to slope-intercept form.
The slope of 4 indicates that for every 1 unit increase in x, the value of y increases by 4 units. This information can be used to find other points on the line, either by moving along the line from the y-intercept or from the point (5, -8) that we identified from the point-slope form.
Understanding the equation's slope and a point on the line is crucial for determining whether other given points lie on the same line. We can use this information to verify the given points in the next section.
Verifying Points on the Line
Now that we have a clear understanding of the equation y + 8 = 4(x - 5), its slope, and a point on the line, we can proceed to verify which of the given points lie on the line. To do this, we will substitute the x and y coordinates of each point into the equation and check if the equation holds true. If the equation is satisfied, the point lies on the line; otherwise, it does not.
Let's test each of the provided points:
A. (4, 5)
Substitute x = 4 and y = 5 into the equation y + 8 = 4(x - 5):
- 5 + 8 = 4(4 - 5)
- 13 = 4(-1)
- 13 = -4
This is not true, so the point (4, 5) does not lie on the line.
B. (-5, -8)
Substitute x = -5 and y = -8 into the equation y + 8 = 4(x - 5):
- -8 + 8 = 4(-5 - 5)
- 0 = 4(-10)
- 0 = -40
This is not true, so the point (-5, -8) does not lie on the line.
C. (5, 0)
Substitute x = 5 and y = 0 into the equation y + 8 = 4(x - 5):
- 0 + 8 = 4(5 - 5)
- 8 = 4(0)
- 8 = 0
This is not true, so the point (5, 0) does not lie on the line.
D. (4, -8)
Substitute x = 4 and y = -8 into the equation y + 8 = 4(x - 5):
- -8 + 8 = 4(4 - 5)
- 0 = 4(-1)
- 0 = -4
This is not true, so the point (4, -8) does not lie on the line.
E. (5, -8)
Substitute x = 5 and y = -8 into the equation y + 8 = 4(x - 5):
- -8 + 8 = 4(5 - 5)
- 0 = 4(0)
- 0 = 0
This is true, so the point (5, -8) does lie on the line.
F. (5, 8)
Substitute x = 5 and y = 8 into the equation y + 8 = 4(x - 5):
- 8 + 8 = 4(5 - 5)
- 16 = 4(0)
- 16 = 0
This is not true, so the point (5, 8) does not lie on the line.
Therefore, after testing all the points, we find that only the point E. (5, -8) satisfies the equation and lies on the line. This confirms our earlier observation from the point-slope form of the equation.
Conclusion: The Point on the Line
In conclusion, we have successfully identified the point that lies on the line described by the equation y + 8 = 4(x - 5). By converting the equation to slope-intercept form and also recognizing its point-slope form, we gained valuable insights into the line's properties, including its slope and a specific point on the line. We then systematically tested each of the provided points by substituting their coordinates into the equation. This process allowed us to determine that only the point (5, -8) satisfies the equation and therefore lies on the line.
This exercise demonstrates the importance of understanding linear equations and their different forms. The ability to manipulate equations, identify key features like slope and intercepts, and verify points on a line are essential skills in mathematics and related fields. By mastering these concepts, individuals can confidently tackle a wide range of problems involving linear relationships.
The application of these skills extends beyond simple equation solving. Linear equations are used extensively in modeling real-world scenarios, making predictions, and optimizing various processes. Whether it's determining the trajectory of a projectile, analyzing financial data, or designing efficient systems, linear equations provide a powerful framework for understanding and solving problems. Therefore, a strong foundation in linear equations is crucial for success in many academic and professional pursuits.
