Problem 45 - Algebra and Trigonometry

Question

Find the slope and \(y\)-intercept of the line and draw its graph. \(\frac{1}{2} x-\frac{1}{3} y+1=0\)

Step-by-Step Solution

Answer

Slope: \(\frac{3}{2}\), y-intercept: 3.

Step 1: Rearrange the equation into slope-intercept form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start with the equation \(\frac{1}{2}x - \frac{1}{3}y + 1 = 0\). First, isolate \(y\) by moving all other terms to the right side of the equation: \(-\frac{1}{3}y = -\frac{1}{2}x - 1\).

Step 2: Solve for y

To solve for \(y\), multiply each term by \(-3\) to eliminate the fraction: \(y = \frac{3}{2}x + 3\). Now the equation is in the form \(y = mx + b\).

Step 3: Identify the slope and y-intercept

In the slope-intercept form \(y = \frac{3}{2}x + 3\), the slope \(m\) is \(\frac{3}{2}\) and the y-intercept \(b\) is \(3\).

Step 4: Draw the graph

To graph the line, start by plotting the y-intercept (0,3) on the y-axis. Then, use the slope \(\frac{3}{2}\), which means move up 3 units and right 2 units from the y-intercept, to find another point on the line. Draw the line through these points.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting a straight line on a coordinate plane. To get started, you need the equation of the line, typically rearranged in slope-intercept form. This form, represented as \(y = mx + b\), clearly reveals the slope \(m\) and the y-intercept \(b\).

To graph the equation, you first pinpoint the y-intercept on the y-axis. This is your starting point. From there, utilize the slope to find other points that lie on the line. For instance, the slope shows you how much to vertically move up or down and horizontally move right or left from one point. By connecting these points with a straight line, you've successfully graphed the equation.

This approach helps in visually understanding the behavior of linear relationships.

Identifying Slope

The slope of a linear equation indicates the steepness and direction of the line. In the slope-intercept form \(y = mx + b\), the slope \(m\) is the coefficient of \(x\). It is expressed as a ratio of the vertical change ('rise') to the horizontal change ('run') between any two points on the line.

• If \(m\) is positive, the line rises as it moves from left to right.
• If \(m\) is negative, the line falls as it moves from left to right.
• If \(m\) is zero, the line is horizontal.

In the equation \(y = \frac{3}{2}x + 3\), the slope \(\frac{3}{2}\) tells us that for every 2 units you move to the right, you should move up by 3 units. Understanding the slope makes it easier to sketch the line without needing multiple calculations.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the equation \(y = mx + b\), the y-intercept is represented by \(b\). It is the value of \(y\) when \(x\) is zero, essentially where the line begins on the vertical axis. This is your initial point when graphing.

For example, in the equation \(y = \frac{3}{2}x + 3\), the y-intercept is \(3\). This means the line crosses the y-axis at the point \((0,3)\). Knowing this starting point makes it easy to draw the line accurately and understand part of the graph's context without needing to calculate further coordinates.

Equation of a Line

The equation of a line is a mathematical representation of all the points that form the line on a graph. The most commonly used form is the slope-intercept form \(y = mx + b\), because it easily shows both the slope and the y-intercept.

This equation helps in quickly determining the line's direction and steepness, and where it crosses the y-axis. If you have a different form of a line's equation, such as \(Ax + By + C = 0\), you can transform it into the slope-intercept form by isolating \(y\) on one side. This is precisely what was done with the equation \(\frac{1}{2}x - \frac{1}{3}y + 1 = 0\), resulting in \(y = \frac{3}{2}x + 3\).

Understanding the equation of a line is crucial for grasping how linear relationships function and how they translate into visual graphs.