Problem 46 - Algebra and Trigonometry

Question

Find the slope and y-intercept of the line, and draw its graph. $$ 3 x-2 y=12 $$

Step-by-Step Solution

Answer

Slope is \(\frac{3}{2}\); y-intercept is \(-6\). Graph by plotting \((0, -6)\) and using the slope.

Step 1: Rearrange the equation to slope-intercept form

The slope-intercept form of a line is given by \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. To rewrite the equation \( 3x - 2y = 12 \) in this form, solve for \( y \). Start by moving \( 3x \) to the right side: \( -2y = -3x + 12 \).

Step 2: Solve for y

Divide every term in the equation \( -2y = -3x + 12 \) by \(-2\) to solve for \( y \). This gives \( y = \frac{3}{2}x - 6 \). Now, the equation is in the form \( y = mx + c \), where \( m = \frac{3}{2} \) and \( c = -6 \).

Step 3: Identify the slope and y-intercept

From the equation \( y = \frac{3}{2}x - 6 \), the slope \( m \) is \( \frac{3}{2} \), and the y-intercept \( c \) is \(-6\).

Step 4: Graph the line

To graph the line \( y = \frac{3}{2}x - 6 \), start by plotting the y-intercept, \( (0, -6) \) on the coordinate plane. Then, use the slope \( \frac{3}{2} \) to find another point. From \( (0, -6) \), move up 3 units and right 2 units to plot another point. Draw a line through these points.

Key Concepts

Slope Calculation

The slope of a line is a crucial concept when working with linear equations. It essentially tells us how steep the line is. The slope, often represented by the letter \( m \), is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. This is usually expressed as the formula:

• \( m = \frac{\text{Rise}}{\text{Run}} \)

To calculate the slope from the equation \( 3x - 2y = 12 \), we first rearranged the equation into the slope-intercept form \( y = mx + c \). Solving gives us \( y = \frac{3}{2}x - 6 \). Here, \( m = \frac{3}{2} \), meaning for every 2 units you move to the right, you move up 3 units. This ratio helps us understand the direction and angle of the line in a coordinate system.

Y-intercept Identification

The y-intercept is the point where the line crosses the y-axis. This value is vital as it provides a starting point for drawing the graph of an equation. It is represented by the letter \( c \) in the slope-intercept form \( y = mx + c \).From our equation \( y = \frac{3}{2}x - 6 \), we identify the y-intercept as \( c = -6 \). This means that when the value of \( x \) is zero, the value of \( y \) is \(-6\). On a graph, this point is plotted at \((0, -6)\). Identifying this point was the first step in our graphing process, as it is where the line begins on the y-axis.

Linear Equations

Linear equations represent straight lines when graphed on a coordinate plane. They are typically written in two forms - standard form and slope-intercept form.

• Standard form: \( Ax + By = C \)
• Slope-intercept form: \( y = mx + c \)

Transforming a linear equation from standard form to slope-intercept form is crucial for easily identifying the slope and y-intercept, aiding in graphing efforts. The original equation given \( 3x - 2y = 12 \) was in standard form. By solving for \( y \), we put it into slope-intercept form, \( y = \frac{3}{2}x - 6 \). This transformation helps us understand and visualize how lines behave on a graph.

Graphing Lines

Graphing lines can be straightforward once you have the slope-intercept form of an equation. This form provides all the necessary details to plot the line on a coordinate plane.To graph our line \( y = \frac{3}{2}x - 6 \):

• Start by plotting the y-intercept \((0, -6)\).
• Use the slope \( \frac{3}{2} \) to find a second point. From \((0, -6)\), move up 3 units and right 2 units, reaching another point.

These points direct us in drawing a straight line through them, representing the equation. Remember, the slope ensures the line's steepness is consistent, providing a visual representation of the linear equation.