Identify An Equation Of A Line In Slope-intercept Form That Passes Through (1, 2) With A Slope Of -8
Understanding how to find the equation of a line is a fundamental skill in algebra and coordinate geometry. When given specific information such as a point through which the line passes and its slope, you can quickly determine the line's equation in slope-intercept form. In this article, we will focus on how to identify an equation of a line in slope-intercept form that passes through the point (1, 2) with a slope of -8. This process involves applying the point-slope form and then converting it into the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
Understanding the Slope-Intercept Form of a Line
Before diving into the specific problem, it's essential to understand what the slope-intercept form of a line entails.
Definition of Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as:- y = mx + b
where:
- m represents the slope of the line
- b represents the y-intercept, or the point where the line crosses the y-axis
This form is particularly useful because it directly gives you the slope and y-intercept, making graphing and understanding the line straightforward.
Importance of the Slope and Point
Knowing the slope and a specific point on the line allows you to quickly write the equation. The key is to use the point-slope form, which is:- y - y₁ = m(x - x₁)
where (x₁, y₁) is a point on the line, and m is the slope.
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Steps to Find the Equation of a Line in Slope-Intercept Form
Let's walk through the process of deriving the equation step by step, given the point (1, 2) and the slope -8.
Step 1: Write the Point-Slope Equation
Using the point-slope form:- Point: (x₁, y₁) = (1, 2)
- Slope: m = -8
Plug these into the formula:
y - 2 = -8(x - 1)
Step 2: Simplify the Equation
Distribute the slope:y - 2 = -8x + 8
Add 2 to both sides to solve for y:
y = -8x + 8 + 2
Simplify:
y = -8x + 10
Step 3: Write the Equation in Slope-Intercept Form
The final form is:- y = -8x + 10
This is the equation of the line passing through (1, 2) with a slope of -8 in slope-intercept form.
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Verifying the Equation of the Line
It's always good practice to verify that your derived equation is correct.
Check if the Point Lies on the Line
Substitute x = 1 into the equation:y = -8(1) + 10 = -8 + 10 = 2
Since y = 2 matches the y-coordinate of the point (1, 2), the point lies on the line, confirming the correctness of the equation.
Graphical Interpretation
Plotting the point (1, 2) and drawing a line with slope -8 will show a line passing through that point and crossing the y-axis at 10, confirming the equation's accuracy.---
Additional Tips for Identifying Line Equations
While the above method works well when you have a point and slope, here are some additional tips:
Using Two Points
If you are given two points, you can:- Calculate the slope using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁)
- Use the point-slope form with either point to find the equation.
Converting from Standard Form
If you are given the line in standard form (Ax + By = C), you can solve for y to convert it into slope-intercept form.Remembering Special Cases
- A line with a slope of 0 is horizontal, with the equation y = constant.
- A vertical line has an undefined slope and can be expressed as x = constant.
Real-World Applications of Line Equations
Understanding how to identify the equation of a line is not only a mathematical exercise but also has practical applications:
- Physics: Modeling constant velocity motion
- Economics: Representing linear cost or revenue functions
- Engineering: Designing components with specific slopes or gradients
- Data Analysis: Fitting lines to data points for trend analysis
Having the ability to quickly determine a line's equation from given data points and slopes is essential in these fields.
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Summary
To summarize, here are the key steps to identify the equation of a line in slope-intercept form passing through a specific point with a given slope:
- Start with the point-slope form: y - y₁ = m(x - x₁).
- Plug in the known point (x₁, y₁) and the slope m.
- Simplify the equation to slope-intercept form y = mx + b.
- Verify the equation by substituting the point into the derived equation.
Applying these steps to the point (1, 2) with a slope of -8 yields the equation:
y = -8x + 10
This precise method ensures accurate and efficient determination of line equations, which is fundamental in algebra, geometry, and various applied sciences.
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Conclusion
Mastering how to identify an equation of a line in slope-intercept form is a cornerstone skill in mathematics. When given a point and a slope, the process involves straightforward steps using the point-slope formula, followed by converting to the slope-intercept form. The example provided illustrates this process clearly, allowing you to apply it to any similar problem. Whether for academic purposes, technical fields, or real-world applications, being able to derive the line's equation accurately enhances your mathematical toolkit and problem-solving abilities.
