Problem 61 - College Algebra

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and \(y\) -intercept. c. Use the slope and y-intercept to graph the linear function. \(2 x+3 y-18-0\)

Step-by-Step Solution

Answer

The slope-intercept form of the given equation is \(y = -2/3 x + 6\), where the slope (m) is -2/3 and the y-intercept (c) is 6.

Step 1: Rewrite the equation in slope-intercept form

The given equation is \(2 x+3 y-18=0\). Rearrange this to make \(y\) the subject, \(3 y = -2 x +18\). Divide each term by 3 to isolate \(y\), giving the equation in the form \(y = mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. The equation becomes \(y = -2/3 x + 6\).

Step 2: Identify the slope and y-intercept

Looking at the equation, the coefficient of \(x\) is the slope, \(m = -2/3\), and the constant term is the y-intercept, \(c = 6\).

Step 3: Graph the linear function

To plot the graph, mark the y-intercept point (0,6) on the graph. The slope is -2/3 meaning for each step to right, go two steps down. Use this process to plot other points on the graph. Connect the points with a straight line to create the graph of the linear function.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a standard way of expressing a linear equation. Its format is written as: \[ y = mx + c \]Here, \(m\) represents the slope of the line, and \(c\) is the y-intercept.

This form is highly useful because it easily reveals both the slope and the y-intercept, making it simple to graph linear equations.

• Convenient for quickly identifying the slope and y-intercept
• Helps in comparing different linear equations

To convert a linear equation to slope-intercept form, isolate \(y\) on one side of the equation, which usually involves some algebraic manipulation.

In the given exercise, the equation \(2x+3y=18\) was rearranged to become \(y = -2/3x + 6\). This transformation makes it easy to see the slope and intercepts.

Slope

The slope of a line is a measure of its steepness and direction. In a slope-intercept equation like \(y = mx + c\), \(m\) stands for the slope.

The slope is the ratio of the change in \(y\) (vertical change) to the change in \(x\) (horizontal change) between any two points on the line.

• Calculates as \(\frac{\text{rise}}{\text{run}}\)
• A positive slope means the line slants upwards; a negative slope means it slants downwards.

In our case, the slope \(m\) is \(-2/3\). This indicates that for every unit increase in \(x\), \(y\) decreases by \(\frac{2}{3}\).

This negative slope results in a declining line when graphed, key for determining direction and steepness.

Y-Intercept

The y-intercept is the point at which a line crosses the y-axis. In the slope-intercept form \(y = mx + c\), the \(c\) value represents the y-intercept.

It is the value of \(y\) when \(x\) is zero, showing where the line will meet the vertical axis.

• Essential for locating the starting point of a line on a graph
• Crucial for setting plots for graphing

In the equation \(y = -2/3x + 6\), the y-intercept is \(6\). This means the line intersects the y-axis at point \((0, 6)\).

Identifying the y-intercept is pivotal for correctly plotting the initial point on the graph.

Graphing Linear Functions

Graphing linear functions involves plotting the line described by the linear equation on a coordinate plane. Understanding the slope and y-intercept is key to this process.

To accurately graph a line:

• Mark the y-intercept on the y-axis.
• Use the slope to find another point. Start at the y-intercept: from here, move according to the slope's rise over run.
• Draw a line through these plotted points.

In our example, the y-intercept \((0, 6)\) is plotted first. Using the slope \(-2/3\), from \( (0, 6) \), go right \(3\) units and down \(2\) units to find the next point.

Connect these points:- Extend the line in both directions for a complete graph.This method works to effectively visualize any linear equation presented in slope-intercept form.