Problem 68 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ m=6, \quad(5,-2) $$

Step-by-Step Solution

Answer

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 1: Identify given information

We are given the slope of the line (m) and a point on the line (x, y): $$ m = 6, \quad (x, y) = (5, -2) $$

Step 2: 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.

Step 3: Substitute given values into the point-slope formula

We substitute the given values into the point-slope formula: $$ y - (-2) = 6(x - 5) $$

Step 4: Simplify the equation

Now, we'll rewrite and simplify the equation: $$ y + 2 = 6(x - 5) $$

Step 5: Expand the equation

Expand the equation by multiplying the slope by x and the constant term: $$ y + 2 = 6x - 30 $$

Step 6: 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: $$y = 6x - 32$$

Key Concepts

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:

• \( y - y_1 = m(x - x_1) \)

Here:

• \( y_1 \) is the y-coordinate of the given point,
• \( x_1 \) is the x-coordinate,
• \( m \) is the slope.

For example, if we're given a slope of 6 and a point (5, -2), substituting these values into the formula gives us:

• \( y + 2 = 6(x - 5) \)

This formula allows us to capture the line's behavior using just one point and its steepness, or slope.

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.

When given specifics like a point and a slope, the transition to different equations can give even better insights. After using the point-slope form, you can convert to the slope-intercept form by solving for \( y \). For example:

• Starting with \( y + 2 = 6(x - 5) \), simplify to \( y = 6x - 32 \).

This finalized slope-intercept equation, \( y = 6x - 32 \), clearly shows both the slope (6) and the y-intercept (-32), making graphing more straightforward.

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':

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of any two distinct points on the line.

This ratio is so vital because:

• 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.

In our problem, a slope of 6 tells you that for every unit increase in \( x \), \( y \) increases by 6 units.

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.

Tools such as the slope, midpoints, and distance formulas all integrate algebra and geometry:

• 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.

Understanding these relationships via coordinate geometry allows solving complex problems by converting them to algebraic equations. Thus, it provides a bridge between visual geometry and analytical algebra, simplifying much of the work involved in analyzing shapes and their relationships.