Problem 23 - College Algebra

Question

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form. \((1,2)\) with a slope of \(-\frac{4}{5}\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{4}{5}x + \frac{14}{5} \).

Step 1: Understanding the Point-Slope Formula

The point-slope form of a line's equation is given by the formula \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a known point on the line and \( m \) is the slope. We are given \( (x_1, y_1) = (1, 2) \) and \( m = -\frac{4}{5} \).

Step 2: Substitute Values into the Formula

Substitute the known values into the point-slope formula: \( y - 2 = -\frac{4}{5}(x - 1) \). This equation incorporates both the given point and the slope.

Step 3: Simplify the Equation

Distribute \(-\frac{4}{5}\) on the right side: \( y - 2 = -\frac{4}{5}x + \frac{4}{5} \). This simplifies the right-hand side of the equation.

Step 4: Convert to Slope-Intercept Form

Rearrange the equation to get \( y \) on one side: Add 2 to both sides, resulting in \( y = -\frac{4}{5}x + \frac{4}{5} + 2 \).

Step 5: Simplify the Slope-Intercept Form

Combine like terms: \( y = -\frac{4}{5}x + \frac{4}{5} + \frac{10}{5} = -\frac{4}{5}x + \frac{14}{5} \). This gives the slope-intercept form, \( y = -\frac{4}{5}x + \frac{14}{5} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most useful equations when it comes to describing a line on a graph. This form is represented as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept, which is where the line crosses the y-axis.
Understanding the structure of this form makes plotting a line straightforward. Once you know \( m \) and \( b \), you can quickly determine the steepness and vertical position of the line.

To convert a line's equation from point-slope form to slope-intercept form, you need to manipulate the equation so that \( y \) is isolated on one side. This often involves rearranging terms and solving for \( y \) as seen in our exercise. Once these steps are mastered, expressing any line in slope-intercept form becomes almost effortless.
Using the slope-intercept form directly lets you quickly sketch graphs and understand relationships between variables in real-world applications.

Equation of a Line

Finding the equation of a line requires understanding two key components: the slope of the line and a point through which it passes. The slope indicates the line's steepness and direction, while a point provides a specific location on the line.
The point-slope formula, \( y - y_1 = m(x - x_1) \), serves as a versatile tool to find this equation. Given a slope \( m \) and a known point \((x_1, y_1)\), you fill in these values to craft the equation of the line.

• **Point**: Indicates a fixed location.
• **Slope**: Describes how much the line rises or falls as you move from left to right.

In the original exercise, we used point \((1, 2)\) and slope \(-\frac{4}{5}\) to derive the line's equation. The ability to swiftly convert the point-slope formula into the more common slope-intercept form is a handy algebraic skill that unites various forms of linear equations.

Algebraic Manipulation

Algebraic manipulation is at the heart of solving equations and finding line forms. It involves a series of mathematical steps used to rearrange and simplify equations.
In our step-by-step solution, algebraic manipulation allowed us to convert the point-slope form into the slope-intercept form. This involved distributing \(-\frac{4}{5}\) in the equation \( y - 2 = -\frac{4}{5}(x - 1) \) and then collecting terms to isolate \( y \).

Key techniques include:

• *Distributive Property*: Applying a multiplier to each term inside parentheses.
• *Combining Like Terms*: Adding or subtracting terms with the same variable parts.
• *Rearranging*: Moving terms to isolate a variable, often making use of inverse operations like adding, subtracting, multiplying, or dividing both sides of an equation.

By following these techniques methodically, one can transform equations while preserving equality, thereby making complex problems more manageable and bring clarity to otherwise intricate algebraic expressions.