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Magazine Chapter 7: Problem 68
Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ m=6, \quad(5,-2) $$
Short Answer
Expert verified
Question: Find the equation of the line with slope 6 and passing through the point (5, -2).
Answer: The equation of the line is $$y = 6x - 32$$.
Step by step solution
01
Identify given information
We are given the slope of the line (m) and a point on the line (x, y):
$$
m = 6, \quad (x, y) = (5, -2)
$$
02
Use the point-slope formula
The point-slope formula is used to find the equation of a line given its slope and a point on the line, and it is given by:
$$
y - y_1 = m (x - x_1)
$$
Where \((x_1, y_1)\) is the given point on the line and m is the slope.
03
Substitute given values into the point-slope formula
We substitute the given values into the point-slope formula:
$$
y - (-2) = 6(x - 5)
$$
04
Simplify the equation
Now, we'll rewrite and simplify the equation:
$$
y + 2 = 6(x - 5)
$$
05
Expand the equation
Expand the equation by multiplying the slope by x and the constant term:
$$
y + 2 = 6x - 30
$$
06
Write the equation in slope-intercept form
To write the equation in slope-intercept form, we need to isolate y on one side of the equation:
$$
y = 6x - 30 - 2
$$
$$
y = 6x - 32
$$
The equation of the line in slope-intercept form is: $$y = 6x - 32$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
point-slope formula
To understand how to find the equation of a line when given a point and a slope, we'll use the **point-slope formula**. This formula is handy because it takes the slope of the line (( m )) and any point on the line (( x_1, y_1 )) to express the line's equation. The formula is written as:
- \( y - y_1 = m(x - x_1) \)
- \( y_1 \) is the y-coordinate of the given point,
- \( x_1 \) is the x-coordinate,
- \( m \) is the slope.
- \( y + 2 = 6(x - 5) \)
equation of a line
Writing the **equation of a line** encapsulates all points on that line. There are multiple forms to express this equation. Among the most useful are the point-slope form and the slope-intercept form. Each form has its applications:
- **Point-Slope Form**: \( y - y_1 = m(x - x_1) \). Useful when you have a slope and a point.
- **Slope-Intercept Form**: \( y = mx + b \). This is preferred when you need to know the slope and the y-intercept directly for graphing purposes.
- Starting with \( y + 2 = 6(x - 5) \), simplify to \( y = 6x - 32 \).
slope
Understanding the **slope** is crucial in coordinate geometry because it measures the steepness or the incline of a line on a graph. The slope, denoted by \( m \), shows the rate of change between the y-coordinates and the x-coordinates. It's calculated as the ratio of the 'rise' over the 'run':
- \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
- \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of any two distinct points on the line.
- A positive slope means the line ascends as you move from left to right.
- A negative slope implies the line descends as you move left to right.
- A zero slope denotes a perfectly horizontal line.
- An undefined slope means a vertical line.
coordinate geometry
**Coordinate geometry** explores the relationship between algebra and geometry using a coordinate plane. Lines and their equations form a core part of this study. Understanding this helps visualize and solve geometric problems algebraically.
- Points are expressed as (x, y) pairs to show their position on the plane.
- Lines are generally described algebraically, such as with slope and equations.
- Slope helps understand the line's direction and steepness.
- Equations link all points along a line.
- Distance formulas and midpoints take algebraic descriptions back to geometric ideas.
