Problem 76 - Algebra 1

Question

GRAPHING LINEAR SYSTEMS Use the graphing method to solve the linear system and describe its solution(s). $$ \begin{aligned} &3 x-2 y=0\\\ &3 x-2 y=-4 \end{aligned} $$

Step-by-Step Solution

Answer

The system of equations has no solution because the lines represented by the equations are parallel.

Step 1: Rearranging the equations into slope-intercept form

To begin with, both equations need to be rearranged into slope-intercept form \(y = mx + b\), where m is the slope and b is the y-intercept. For the first equation, dividing all terms by -2 gives \(y = 1.5x\). For the second, adding \(3x\) to both sides and dividing by -2 gives \(y = 1.5x + 2\).

Step 2: Graphing the equations

Now both equations can be graphed in the same coordinate plane. The equation \(y = 1.5x\) is a direct variation with a slope of 1.5 or \(3/2\) and passes through the origin. The equation \(y = 1.5x + 2\) is a linear equation with the same slope and y-intercept at 2. The graphs would therefore be two lines with the same slopes but different y-intercepts.

Step 3: Finding the solution

Finally, since the lines don’t cross (they are parallel), the system doesn’t have a solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a powerful tool in algebra for graphing lines with ease. This form is written as \( y = mx + b \), where:

• \( m \) represents the slope, which indicates how steep the line is.
• \( b \) denotes the y-intercept, the point where the line crosses the y-axis.

Understanding the slope-intercept form makes it easier to quickly sketch the graph of a line. The slope \( m \) can be seen as a ratio \( \frac{rise}{run} \), which describes how many units you move vertically (rise) for a specific horizontal movement (run).
To convert a linear equation to this format, rearrange the equation so \( y \) is isolated on one side. For example, in the exercise, the first equation rearranged becomes \( y = 1.5x \), indicating a slope of 1.5 and a y-intercept of 0. The second equation modifies to \( y = 1.5x + 2 \), also with a slope of 1.5 but a y-intercept of 2.

Parallel Lines

In the context of graphing linear systems, the notion of parallel lines is crucial. Lines in the plane are parallel if they have identical slopes but distinct y-intercepts. Parallel lines never intersect or meet; hence, they extend forever in the same direction, maintaining a constant distance apart.
In our exercise, both equations \( y = 1.5x \) and \( y = 1.5x + 2 \) share the same slope of 1.5, indicating that they are parallel lines. However, the first line passes through the origin with a y-intercept of 0, while the second line crosses the y-axis at point \( (0, 2) \).
Understanding parallelism is essential when analyzing systems of equations. If the slopes are the same and the intercepts differ, the lines are parallel and do not intersect, leading to no common solution for the system.

No Solution

A system of equations is said to have no solution when the equations represent parallel lines. This occurs because parallel lines, as mentioned, do not meet at any point.

• When graphing, you'll observe that these lines never intersect with one another.
• No intersection point implies no common solution exists that satisfies both equations simultaneously.

In solving for linear systems, identifying parallel lines immediately signals that the system has no solution. For our original problem, the equations \( y = 1.5x \) and \( y = 1.5x + 2 \) show this scenario. Since they share the same slope but have different y-intercepts, the lines will run parallel without crossing.
Recognizing when a system has no solution helps avoid further unnecessary calculations and allows for a better understanding of the relationship between the equations.