What Is The Equation Of The Line That Has A Slope Of 3 And Passes Through The Point (1, -2)
Understanding how to find the equation of a line given certain parameters is a fundamental aspect of coordinate geometry. When a line's slope and a point through which it passes are known, it becomes straightforward to derive its algebraic equation. In this article, we will explore step-by-step how to determine the equation of a line with a slope of 3 that passes through the point (1, -2). We will delve into the underlying concepts, formulas, and calculations involved, ensuring a comprehensive understanding of this common problem in mathematics.
---
Understanding the Basics: What Is a Line's Equation?
The General Form of a Line Equation
A straight line in a two-dimensional coordinate plane can be represented mathematically by an equation. The most common form is the slope-intercept form:
\[ y = mx + b \]
where:
- \( y \) and \( x \) are the variables representing points on the line,
- \( m \) is the slope of the line,
- \( b \) is the y-intercept, the point where the line crosses the y-axis.
This form is especially useful because it clearly shows the slope and the y-intercept, allowing for quick graphing and analysis.
The Role of Slope and a Point in Defining a Line
The slope (\( m \)) indicates the steepness and direction of the line:
- A positive slope (like 3) means the line ascends from left to right.
- The point through which the line passes provides a specific location on the plane, anchoring the line's position.
When both the slope and a point are known, the goal is to find the specific equation of the line that satisfies these conditions.
---
Deriving the Equation of the Line: Step-by-Step Process
Using the Point-Slope Form
The point-slope form of a line's equation is particularly useful when you know:
- The slope \( m \),
- A point \( (x1, y1) \) that lies on the line.
The formula is:
\[ y - y1 = m(x - x1) \]
This form directly incorporates the known point and slope, simplifying the process of deriving the line's equation.
Applying the Given Data
Given:
- Slope \( m = 3 \),
- Point \( (x1, y1) = (1, -2) \).
Plugging these into the point-slope form:
\[ y - (-2) = 3(x - 1) \]
which simplifies to:
\[ y + 2 = 3(x - 1) \]
---
Converting to Slope-Intercept Form
Step-by-Step Conversion
Starting with:
\[ y + 2 = 3(x - 1) \]
Distribute the 3:
\[ y + 2 = 3x - 3 \]
Subtract 2 from both sides to isolate \( y \):
\[ y = 3x - 3 - 2 \]
Simplify:
\[ y = 3x - 5 \]
This is the slope-intercept form of the line:
\[ \boxed{ y = 3x - 5 } \]
which clearly indicates:
- The slope \( m = 3 \),
- The y-intercept \( b = -5 \).
---
Verifying the Equation
Checking if the Point Lies on the Line
To verify, substitute \( x = 1 \) into the derived equation:
\[ y = 3(1) - 5 = 3 - 5 = -2 \]
which matches the original y-coordinate of the point \( (1, -2) \). Thus, the point indeed lies on the line, confirming that the equation is correct.
Graphical Interpretation
Plotting the line:
- It passes through \( (1, -2) \),
- Has a slope of 3, meaning for each unit increase in \( x \), \( y \) increases by 3 units,
- Crosses the y-axis at \( y = -5 \).
---
Additional Concepts and Variations
Alternate Forms of the Line Equation
While the slope-intercept form is most straightforward, other forms include:
- Standard Form: \( Ax + By = C \)
- Point-Slope Form: \( y - y1 = m(x - x1) \), which we already used.
Converting the derived equation to standard form:
\[ y = 3x - 5 \]
Subtract \( 3x \) from both sides:
\[ -3x + y = -5 \]
or
\[ 3x - y = 5 \]
Both representations are valid and useful in different contexts.
Understanding Line Properties from the Equation
From the equation \( y = 3x - 5 \):
- Slope (\( m \)): 3, indicating the line rises 3 units vertically for every 1 unit horizontally.
- Y-intercept (\( b \)): -5, the point where the line crosses the y-axis.
This information allows for quick sketching and analysis of the line's behavior.
---
Practical Applications and Examples
Example 1: Graphing the Line
To graph the line:
- Plot the y-intercept \( (0, -5) \).
- From this point, use the slope to find another point: move 1 unit right (positive x direction), and 3 units up (since slope is 3), landing at \( (1, -2) \).
- Draw a straight line through these points, extending in both directions.
Example 2: Finding the Intersection with Another Line
Suppose you have another line: \( y = -x + 4 \). To find the intersection:
Set the two equations equal:
\[ 3x - 5 = -x + 4 \]
Solve for \( x \):
\[ 3x + x = 4 + 5 \]
\[ 4x = 9 \]
\[ x = \frac{9}{4} \]
Substitute back into one of the equations:
\[ y = 3 \times \frac{9}{4} - 5 = \frac{27}{4} - 5 = \frac{27}{4} - \frac{20}{4} = \frac{7}{4} \]
The intersection point is:
\[ \left( \frac{9}{4}, \frac{7}{4} \right) \]
---
Conclusion
Finding the equation of a line given a slope and a point involves understanding the fundamental forms of linear equations and applying the appropriate formulas. In this case, with a slope of 3 and passing through the point (1, -2), the process is straightforward:
- Use the point-slope form:
\[ y - y1 = m(x - x1) \]
- Substitute the known values:
\[ y + 2 = 3(x - 1) \]
- Simplify to slope-intercept form:
\[ y = 3x - 5 \]
This equation encapsulates all the information about the line's steepness and position in the coordinate plane. Mastery of this derivation process is crucial for solving a wide range of problems in algebra, coordinate geometry, and related fields. Whether for graphing, analyzing intersections, or applying real-world models, understanding how to derive and manipulate line equations is an essential skill in mathematics.
