Problem 56 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Slope \(-\frac{3}{5},\) through (4,4)

Step-by-Step Solution

Answer

The equation is \(y = -\frac{3}{5}x + \frac{32}{5}\).

Step 1: Understanding Slope-Intercept Form

The slope-intercept form of a line's equation is given by the formula \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept of the line.

Step 2: Substitute the Known Slope

The slope of the line is given as \(-\frac{3}{5}\). So, substitute \(m = -\frac{3}{5}\) into the slope-intercept form: \(y = -\frac{3}{5}x + b\).

Step 3: Use the Point to Find the Y-Intercept

You are given that the point \( (4, 4) \) is on the line. Substitute \(x = 4\) and \(y = 4\) into the equation: \[4 = -\frac{3}{5}(4) + b\]Solve for \(b\):\[4 = -\frac{12}{5} + b\]\[4 + \frac{12}{5} = b\] Convert 4 to a fraction: \[\frac{20}{5} + \frac{12}{5} = b\] \[b = \frac{32}{5}\]

Step 4: Write the Final Equation

Now that you know both \(m\) and \(b\), substitute them into the slope-intercept form to get the equation of the line: \(y = -\frac{3}{5}x + \frac{32}{5}\).

Key Concepts

Equation of a Line

In mathematics, the equation of a line represents a straight path that extends infinitely in both directions within the coordinate system. The most common way to express this is using the slope-intercept form, which is expressed as \(y = mx + b\). Here, \(m\) denotes the slope of the line, and \(b\) indicates the y-intercept.
This form is particularly useful because it simplifies the process of graphing and solving linear equations. By rewriting a line's equation into this form, we can easily identify the slope and y-intercept, making it straightforward to understand the behavior and position of the line on a graph.

Slope

The slope of a line, symbolized by \(m\) in the equation \(y = mx + b\), measures the steepness or incline of the line. It is calculated as the change in the y-coordinates divided by the change in the x-coordinates between two distinct points on the line. Mathematically, this can be expressed using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
where \((x_1, y_1)\) and \((x_2, y_2)\) are points on the line.
A positive slope means the line ascends from left to right, while a negative slope implies it descends. A zero slope results in a horizontal line, and an undefined slope results in a vertical line.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis, represented by \(b\) in the slope-intercept form \(y = mx + b\). It's the value of \(y\) when \(x\) equals zero. The y-intercept provides a starting point for drawing the line on a graph, helping in determining its position relative to the origin.
For example, if you know a line has a slope of \(-\frac{3}{5}\) and passes through the point \((4, 4)\), you can calculate the y-intercept by substituting these values into the slope-intercept formula. This way, you can determine the full equation of the line, including where it touches the y-axis.

Coordinate System

A coordinate system is a framework that uses numbers to uniquely define the position of a point in space. In the 2D coordinate system, a pair of numerical values, \(x\) and \(y\), describe the location of a point in a plane. These coordinates are represented as ordered pairs \((x, y)\) and plotted within a grid defined by the x-axis (horizontal) and y-axis (vertical).
This system allows for accurate placement and analysis of points and lines, including understanding graphs such as lines represented by their equations. When we talk about a line passing through a specific point, such as \((4,4)\), we use the coordinate system to precisely plot its position within the grid.