Problem 69 - Algebra 1

Question

Decide whether the graphs of the two equations are $$ 3 x+9 y+2=0 ; 2 y=-6 x+3 $$

Step-by-Step Solution

Answer

No, the graphs of the two equations are not the same.

Step 1: Simplify the first equation

The first equation is \(3x + 9y + 2 = 0\). To simplify this equation into slope-intercept form \(y=mx+b\), first isolate y. The first step is to subtract \(3x + 2\) from both sides, which results in \(9y = -3x - 2\). Divide every term by 9 to get \(y = -\frac{1}{3}x - \frac{2}{9}\).

Step 2: Simplify the second equation

The second equation is \(2y = -6x + 3\). To simplify this equation into slope-intercept form \(y=mx+b\), divide every term by 2 to get \(y = -3x + \frac{3}{2}\).

Step 3: Compare the two equations

From the above simplifications, first equation is \(y = -\frac{1}{3}x - \frac{2}{9}\) and the second equation is \(y = -3x + \frac{3}{2}\). Compare the slopes and the y-intercepts of the two equations. The slope of the first equation is -\frac{1}{3} and the slope of the second equation is -3. Similarly, the y-intercept of the first equation is -\frac{2}{9} and the y-intercept of the second equation is \frac{3}{2}. Since both the slopes and the y-intercepts are not equal, the given equations are not the same.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is key to working with linear equations. It is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. This equation is particularly useful because it provides two important characteristics of the line directly within its format.

When you see an equation, converting it into slope-intercept form allows you to instantly understand these characteristics:

• Slope (\( m \)): This indicates the steepness and direction of the line. A positive slope means the line is rising from left to right, while a negative slope means it is falling.
• Y-intercept (\( b \)): This is the point where the line crosses the y-axis. It tells us the value of \( y \) when \( x \) is zero.

To transform a general linear equation into slope-intercept form, you isolate \( y \) on one side of the equation. For example, in the equation \( 3x + 9y + 2 = 0 \), you rearrange it by isolating \( y \) to achieve \( y = -\frac{1}{3}x - \frac{2}{9} \). This shows the slope is \(-\frac{1}{3}\) and the y-intercept is \(-\frac{2}{9}\). This understanding forms the basis for graphing and comparing equations.

Graphing Equations

Once you have a linear equation in slope-intercept form, graphing becomes much easier. The form itself provides a clear guide on where to start and how to proceed. The process involves plotting the y-intercept and using the slope to determine another point on the line.

Here's how you can do it effectively:

• Start with the y-intercept: Place your first point at the y-intercept. For instance, with \( y = -\frac{1}{3}x - \frac{2}{9} \), your initial point is on \( y = -\frac{2}{9} \), near the origin.
• Use the slope to find the next point: The slope \( -\frac{1}{3} \) tells you that for every step 3 units to the right on the x-axis, you move 1 unit down on the y-axis. Mark this second point from your y-intercept, then draw a line through these points.

By following these steps, you can accurately draw the graph of any linear equation. Graphing is an intuitive way to visualize equations and better understand the differences and similarities between them.

Linear Equations

Linear equations represent straight lines on a graph. They are essential in algebra as they model relationships with constant rates of change and are foundational for more complex topics. In its simplest form, a linear equation can be written in the standard forms such as \( ax + by = c \) or the more convenient slope-intercept form \( y = mx + b \).

Linear equations are powerful because they:

• Express constant relationships: The slope signifies this constant rate of change—how much \( y \) changes for a change in \( x \).
• Are predictable: Given any \( x \), you can readily calculate its corresponding \( y \).
• Offer insights visually: When graphed, linear equations provide an immediate understanding of relations, intersections, and parallelism.

With the knowledge of linear equations, you're better equipped to analyze and interpret data, making it a fundamental concept in both mathematics and real-world applications.