Problem 40 - Algebra 1: Concepts and Skills

Question

Write the equation in slope-intercept form. Then graph the equation. $$2 x-3 y-6=0$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = \frac{2}{3}x - 2\). For the graph, plot the y-intercept (-2) on the y-axis, move up 2 units and right 3 units, and draw a line through these points.

Step 1: Transform the Equation to Slope-Intercept Form

To change the given equation \(2x - 3y - 6 = 0\) into the slope-intercept form, isolate \(y\) on one side of the equation. This can be achieved by firstly moving \(2x\) and \(6\) to the other side so that the equation becomes \(-3y = -2x + 6\). Then, division each side by \(-3\) gives the equation in slope-intercept form \(y = \frac{2}{3}x - 2\).

Step 2: Determine the slope and the y-intercept

From the slope-intercept form \(y = \frac{2}{3}x - 2\), the slope \(m\) is \(\frac{2}{3}\) and the y-intercept \(b\) is \(-2\). The slope and y-intercept are used to graph the line.

Step 3: Graph the Equation

To draw the graph of the equation: 1. Plot the y-intercept which is -2 on the y-axis. 2. From here, move up 2 units (the rise) and to the right 3 units (the run) as directed by the slope \(\frac{2}{3}\). 3. Finally, draw a line through the two points.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves visually representing a linear equation on a coordinate plane. The equation is typically in slope-intercept form, represented as \( y = mx + b \), where \( m \) denotes the slope of the line and \( b \) is the y-intercept. This form makes it straightforward to sketch a graph.When graphing:

• You start by identifying the y-intercept (\( b \)), which is the point where the line crosses the y-axis. This point will have coordinates \((0, b)\).
• The slope \( m \) dictates the direction and steepness of the line. It is the ratio of the vertical change (the rise) to the horizontal change (the run) between two points on the line.
• After plotting the y-intercept, use the slope to find another point on the line. This helps in ensuring that the graph is accurate.
• Connect the plotted points with a straight line stretching in both directions.

Graphing is a visual tool that helps understand the properties and relationships covered by the linear equation, such as intercepts and gradients. It offers a clear way to check solutions to linear problems by showing visually whether points align with the equation.

Slope

The concept of slope is central to understanding linear equations. The slope of a line describes its steepness, incline, or gradient. In the slope-intercept form \( y = mx + b \), the slope \( m \) can be found as the coefficient in front of \( x \).The slope is calculated as:

• \( m = \frac{\text{rise}}{\text{run}} \)
• "Rise" is the vertical change and "run" is the horizontal change between two points on a line.

For example, a slope of \( \frac{2}{3} \) means that for every 3 units you move horizontally to the right, you move 2 units up vertically. This ratio is consistent across the entire line.A positive slope indicates an upward, rightward direction, showing that as one variable increases, the other does too. Conversely, a negative slope would fall to the right, indicating an inverse relationship. A slope of zero depicts a horizontal line, while an undefined slope indicates a vertical line.Understanding slope helps in predicting the behavior of linear charts and functions and is essential in problems involving rates and trends.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis of the coordinate plane. In the slope-intercept form \( y = mx + b \), it is represented by \( b \).Identifying the y-intercept:

• The y-intercept is always a point on the graph where \( x = 0 \).
• The coordinates of the y-intercept are \( (0, b) \).
• For example, if \( b = -2 \), then the line crosses the y-axis at the point \( (0, -2) \).

The y-intercept provides a starting point for graphing a line. From this point, you apply the slope to find other points and draw the full line.In real-world contexts, the y-intercept often represents a starting value or initial condition when measuring change. For instance, in a financial example, the y-intercept might represent an initial cost before any product is sold.