Problem 6 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((-1,-4)\) with slope \(-\frac{4}{5}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the point-slope equation form

The point-slope form of a line's equation is given by \[ y - y_1 = m(x - x_1) \] where \(x_1, y_1\) is a point on the line and \(m\) is the slope.

Step 2: Substitute the given point and slope

Plug in the given point \((-1, -4)\) and the slope \(-\frac{4}{5}\). This gives us \[ y - (-4) = -\frac{4}{5}(x - (-1)) \] simplifying to \[ y + 4 = -\frac{4}{5}(x + 1) \]

Step 3: Simplify the equation

Distribute the slope \[ y + 4 = -\frac{4}{5}x - \frac{4}{5} \] Subtract 4 from both sides to solve for \(y\) \[ y = -\frac{4}{5}x - \frac{4}{5} - 4 \] Convert the constants to have a common denominator \[ y = -\frac{4}{5}x - \frac{4}{5} - \frac{20}{5} \] simplifying to \[ y = -\frac{4}{5}x - 5 \]

Key Concepts

Point-Slope Form

The point-slope form of a line's equation is used when you know a single point on a line and the slope of the line. It is a valuable tool, especially when beginning the exploration of linear equations. The standard formula for point-slope form is: \[ y - y_1 = m(x - x_1) \] Here,

• \(y_1\) refers to the y-coordinate of the known point on the line.
• \(x_1\) is the x-coordinate of the same point.
• \(m\) is the slope of the line.

By substituting the coordinates of the point and the slope into this equation, you establish a linear relationship specific to your data. For example, given the point \((-1, -4)\) with a slope of \(-\frac{4}{5}\), substituting into the formula gives: \[ y - (-4) = -\frac{4}{5}(x - (-1)) \], which simplifies to \[ y + 4 = -\frac{4}{5}(x + 1) \].

Slope

The slope of a line measures its steepness. In mathematical terms, it represents the change in y divided by the change in x between any two points on the line. The slope is denoted by \(m\) and can be calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In our example, the slope is given directly as \(-\frac{4}{5}\). This means for every 5 units the line moves horizontally, it moves down 4 units vertically. Visualizing the slope helps in understanding the distribution of points along the line. When you substitute the slope into the point-slope form, it controls how steeply the line rises or falls. Understanding how the slope affects the line's direction is crucial for graphing and analyzing linear equations.

Simplifying Equations

Simplifying equations involves breaking down expressions to their simplest form for easier interpretation and analysis. This is particularly useful when you need to rewrite point-slope form into slope-intercept form (\(y = mx + b\)). Here's how we simplify the given example: 1. Start with inserting the point and the slope into the point-slope form: \[ y + 4 = -\frac{4}{5}(x + 1) \] 2. Distribute the slope: \[ y + 4 = -\frac{4}{5}x - \frac{4}{5} \] 3. Isolate \(y\) by subtracting 4 from both sides: \[ y = -\frac{4}{5}x - \frac{4}{5} - 4 \] 4. Convert constants to a common denominator: \[ y = -\frac{4}{5}x - \frac{4}{5} - \frac{20}{5} \] 5. Subtract the constants to reach the simplest form: \[ y = -\frac{4}{5}x - 5 \] Simplifying equations not only makes them more readable but also aids in graphing and understanding the linear relationship more clearly.