Equation Of A Line With Slope 10 Through (8, -2) In Slope-Intercept And Point-Slope Form

In the realm of coordinate geometry, one of the fundamental concepts is determining the equation of a straight line. A line can be uniquely defined if we know its slope and a point that lies on it. In this article, we will explore how to find the equation of a line given its slope and a point, specifically when the slope is 10 and the point is (8, -2). We will express the equation in two common forms: slope-intercept form and point-slope form. Understanding these forms is crucial for various applications in mathematics, physics, engineering, and computer graphics.

Understanding Slope-Intercept Form

The slope-intercept form is a widely used way to represent the equation of a line. It is expressed as:

y = mx + b

where:

• y is the dependent variable (typically the vertical coordinate).
• m is the slope of the line, representing the rate of change of y with respect to x. A slope of 10 indicates that for every 1 unit increase in x, y increases by 10 units.
• x is the independent variable (typically the horizontal coordinate).
• b is the y-intercept, which is the point where the line intersects the y-axis (i.e., the value of y when x is 0). The y-intercept is a critical parameter as it anchors the line's position on the coordinate plane.

To find the equation of the line in slope-intercept form, we need to determine the values of m and b. We are given the slope m = 10 and a point (8, -2) that the line passes through. We can use this information to solve for b. The process involves substituting the given slope and the coordinates of the point into the slope-intercept equation and solving for the y-intercept. This method is a direct application of the definition of a line and its slope, making it a fundamental technique in linear algebra.

Finding the Y-Intercept (b)

We know that the line passes through the point (8, -2), which means that when x = 8, y = -2. We also know that the slope m = 10. Plugging these values into the slope-intercept form equation, we get:

-2 = 10(8) + b

This equation can be simplified to solve for b. This step is crucial as it determines the line's vertical positioning on the coordinate plane. The y-intercept acts as the anchor point for the line, and its correct calculation is essential for the line's accurate representation.

-2 = 80 + b

To isolate b, we subtract 80 from both sides of the equation:

b = -2 - 80
b = -82

Thus, the y-intercept b is -82. This value tells us that the line intersects the y-axis at the point (0, -82). Knowing the y-intercept and the slope, we can now write the complete slope-intercept form of the equation.

Writing the Equation in Slope-Intercept Form

Now that we have found the slope m = 10 and the y-intercept b = -82, we can substitute these values into the slope-intercept form equation y = mx + b:

y = 10x - 82

This is the equation of the line in slope-intercept form. It clearly shows the relationship between y and x, with the slope indicating the rate of change and the y-intercept indicating the line's vertical position. The slope-intercept form is particularly useful for quickly identifying the slope and y-intercept of a line, making it a standard form in linear equations.

Understanding Point-Slope Form

The point-slope form is another way to represent the equation of a line. It is particularly useful when you know a point on the line and the slope, but not necessarily the y-intercept. The point-slope form is expressed as:

y - y₁ = m(x - x₁)

where:

• y is the dependent variable.
• y₁ is the y-coordinate of a known point on the line.
• m is the slope of the line.
• x is the independent variable.
• x₁ is the x-coordinate of the same known point on the line.

The point-slope form directly incorporates the slope and a specific point on the line, making it an intuitive way to define a line. It highlights the relationship between the change in y and the change in x relative to a fixed point. This form is especially advantageous when dealing with geometric problems or when transforming between different forms of linear equations.

Using the Point-Slope Form

To find the equation of the line in point-slope form, we use the given point (8, -2) and the slope m = 10. We substitute these values into the point-slope form equation:

y - y₁ = m(x - x₁)

Plugging in the values, we get:

y - (-2) = 10(x - 8)

Simplifying the equation, we have:

y + 2 = 10(x - 8)

This is the equation of the line in point-slope form. It represents the same line as the slope-intercept form but expresses it in a different format. The point-slope form is particularly useful for understanding how the line's equation is constructed from its slope and a specific point. It is also a valuable stepping stone for converting to other forms, such as the slope-intercept form or the standard form of a linear equation.

Converting Between Forms

It's important to understand that both the slope-intercept form and the point-slope form represent the same line, just in different ways. We can convert between these forms using algebraic manipulation. This skill is essential for solving various problems in linear algebra and calculus.

Converting from Point-Slope to Slope-Intercept Form

To convert from point-slope form to slope-intercept form, we need to isolate y on one side of the equation. Starting with the point-slope form we found:

y + 2 = 10(x - 8)

First, distribute the 10 on the right side of the equation:

y + 2 = 10x - 80

Next, subtract 2 from both sides to isolate y:

y = 10x - 80 - 2
y = 10x - 82

This is the same slope-intercept form equation we found earlier, y = 10x - 82. This conversion demonstrates the equivalence of the two forms and how they can be interchanged to suit different problem-solving needs.

Converting from Slope-Intercept to Point-Slope Form

To convert from slope-intercept form to point-slope form, we can use the given point and slope to construct the point-slope equation directly. Starting with the slope-intercept form we found:

y = 10x - 82

We know the slope m = 10 and the point (8, -2). We can substitute these values into the point-slope form equation:

y - y₁ = m(x - x₁)

y - (-2) = 10(x - 8)
y + 2 = 10(x - 8)

This is the same point-slope form equation we found earlier. This conversion reinforces the idea that different forms of linear equations can represent the same line and that understanding these forms allows for flexibility in problem-solving.

Conclusion

In this article, we successfully found the equation of the line with a slope of 10 that passes through the point (8, -2) in both slope-intercept form and point-slope form. The slope-intercept form is y = 10x - 82, and the point-slope form is y + 2 = 10(x - 8). We also demonstrated how to convert between these two forms. Understanding these concepts is fundamental to mastering linear equations and their applications in various fields. The ability to manipulate and interpret different forms of linear equations is a crucial skill for anyone working with mathematical models and real-world applications.

By understanding the slope-intercept and point-slope forms, you gain a powerful tool for analyzing and describing linear relationships. These forms not only provide a concise representation of a line but also offer valuable insights into its properties, such as its steepness and position on the coordinate plane. Mastering these concepts is a significant step towards a deeper understanding of mathematics and its applications.