Problem 51 - Algebra 1

Question

Find the slope and the \(y\) -intercept of the graph of the equation. Then graph the equation. $$ -2 y-3 x=6 $$

Step-by-Step Solution

Answer

The slope of the line is -1.5 and the y-intercept is -3. In the graphical representation, the line passes through the point (0, -3) on the y-axis and proceeds downward as the slope is negative.

Step 1: Rewrite the equation to slope-intercept form

To convert \(-2y - 3x = 6\) into slope-intercept form, first solve for \(y\). First, rearrange the equation to get \(-2y = 3x + 6\). Divide every term by -2 to isolate \(y\). So, \(y = -1.5x -3\).

Step 2: Identify the slope and y-intercept

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Comparing this to our given equation, we can see that \(m = -1.5\) and \(b = -3\). So, the slope of the line is -1.5 and the y-intercept is -3.

Step 3: Graph the equation

Begin by marking the y-intercept (0, -3) on the y-axis. Since the slope is -1.5, move one unit down and 1.5 units to the right from the y-intercept to get the next point (1.5, -4). Draw a line through these two points to graph the equation.

Key Concepts

Linear equations

A linear equation is a type of algebraic equation where the highest power of the variable is one. These equations are called "linear" because they graph as straight lines on a coordinate plane. Linear equations can generally be written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.Linear equations have some key features:

• They produce straight lines when graphed.
• They have both a slope and an intercept, which are key to understanding their graphs.
• Each solution of a linear equation corresponds to a point on its line.

A solution to a linear equation involves finding the value of the variable that makes the equation true. When working with linear equations, converting them into slope-intercept form can make analyzing and graphing them much simpler.

Graphing linear equations

Graphing linear equations is a method of representing these equations visually. By plotting points on a graph, you can see the relationship between the variables \( x \) and \( y \). The graph of a linear equation is always a straight line because of the constant rate of change represented by the line's slope.To graph a linear equation:

• First convert the equation into slope-intercept form, \( y = mx + b \), for easier graphing.
• Identify the \( y \)-intercept (the point where the line crosses the y-axis). This will be the starting point.
• Use the slope to determine the direction and steepness of the line. The slope tells you how many units to rise or fall for each unit moved to the right on the graph.

Once you plot the intercept and use the slope to find another point, draw a line through these points to complete the graph. Understanding how to visualize linear equations through graphing is vital for recognizing the relationship of variables.

Slope and y-intercept

The slope and \( y \)-intercept are the two key components of the slope-intercept form of a linear equation, given by \( y = mx + b \). Each part of this form gives specific information about the line:- **The slope (\( m \))**: This indicates the steepness of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. The magnitude of the slope reflects how sharply the line inclines or declines.- **The \( y \)-intercept (\( b \))**: This is the point where the line crosses the \( y \)-axis. It provides a starting point for graphing the line and helps in visualizing the equation.For the equation \( y = -1.5x - 3 \):

• The slope \( m \) is \(-1.5\), which means for every unit increase in \( x \), \( y \) decreases by 1.5 units.
• The \( y \)-intercept \( b \) is \(-3\), indicating that the line crosses the \( y \)-axis at the point \( (0, -3) \).

Knowing both the slope and \( y \)-intercept allows you to graph the equation efficiently and understand how changes in \( x \) affect \( y \).