Problem 78 - Algebra 1: Concepts and Skills

Question

Find the slope and \(y\) -intercept of the graph of the equation. (Lesson 4.7) $$ 2 x+3 y=6 $$

Step-by-Step Solution

Answer

The slope of the line represented by the equation, \(2x + 3y = 6\), is \(-\frac{2}{3}\) and the y-intercept is \(2\).

Step 1: Rearrange into slope intercept form

The first step will involve rearranging the given equation \(2x + 3y = 6\), to the slope-intercept form \(y = mx + b\). To achieve this, subtract \(2x\) from both sides of the equation. This would give \(3y = 6 - 2x \). Then, divide each term by \(3\) to isolate \(y\). The resulting equation would be \(y = 2 - \frac{2}{3}x \).

Step 2: Identify the slope and y-intercept

After rearranging the equation to slope-intercept form, identify the slope and y-intercept. In the equation, \(y = 2 - \frac{2}{3}x\), the coefficient of \(x\) is the slope, and the constant term is the y-intercept. Therefore, the slope, \(m\) = \(-\frac{2}{3}\), and the y-intercept, \(b\) = \(2\).

Step 3: Rewrite and Solve

Interpret the results. The slope \(m\) = \(-\frac{2}{3}\) means the line decreases as we move along the x-axis. This is because for every 3 units increase in \(x\), there is a decrease of 2 units in \(y\). Also, the y-intercept \(b\) = \(2\) indicates that the line crosses the y-axis at the point (0,2).

Key Concepts

Understanding Slope

In the context of linear equations, the **slope** is a number that represents the steepness and direction of a line on a graph. It is often denoted by the letter \(m\) and is found in the equation of a line when written in slope-intercept form \( y = mx + b \). The slope is calculated by the change in the y-coordinates divided by the change in the x-coordinates between two points on the line.

When interpreting the slope:

• A positive slope means the line ascends from left to right, indicating growth.
• A negative slope means the line descends from left to right, indicating a decline. This is the situation in the example where the slope \(-\frac{2}{3}\) indicates the line is descending.
• A zero slope implies a horizontal line, showing no change as \(x\) changes.
• An undefined slope corresponds to a vertical line, as the change in \(x\) would be zero.

In practical terms, the slope \(-\frac{2}{3}\) means that for every increase of 3 units in \(x\), \(y\) decreases by 2 units.

Y-Intercept: Where the Line Meets the Y-Axis

The **y-intercept** of a line is the point where the line intersects the y-axis. In the slope-intercept form of a linear equation, \(y = mx + b\), the y-intercept is represented by \(b\). It gives a clear starting point of the line on a graph.

This value tells you:

• The point \( (0, b) \) - when the value of \(x\) is zero, the coordinate becomes \((0, b)\).
• In the example given, the y-intercept is \(2\), thus the line crosses the y-axis at the point \( (0, 2) \).

Understanding the y-intercept is crucial because it provides you with the initial or starting value in some real-world scenarios.

Linear Equations and the Slope-Intercept Form

**Linear equations** are mathematical expressions that describe a straight line on a coordinate plane. One of the most common forms of linear equations is the slope-intercept form, \(y = mx + b\). This format allows you to easily identify both the slope and y-intercept:

• \(m\) represents the slope, or how steep the line is.
• \(b\) is the y-intercept, showing where the line crosses the y-axis.

To convert any linear equation into this form, you need to solve for \(y\). The given problem is a perfect example: the original equation \(2x + 3y = 6\) is transformed into \(y = 2 - \frac{2}{3}x\) by manipulating it algebraically.

By writing an equation in slope-intercept form, you easily position a line on a graph, helping determine both its direction and where it starts. This form is extremely useful in graphing and interpreting linear relationships, making it a cornerstone in the study of algebra.