Problem 41 - Algebra A Combined Function

Question

Without graphing, decide. a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point? b. How many solutions does the system have? See Examples 7 and 8 . \(\left\\{\begin{array}{l}2 x+y=0 \\ 2 y=6-4 x\end{array}\right.\)

Step-by-Step Solution

Answer

The lines are parallel with no solutions.

Step 1: Express Both Equations in Slope-Intercept Form

First, we need to write both equations in the slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.- For the first equation, \(2x + y = 0\), we solve for \(y\): \[ y = -2x \] - For the second equation, \(2y = 6 - 4x\), we also solve for \(y\): \[ y = -2x + 3 \] Now we have both equations in slope-intercept form: \(y = -2x\) and \(y = -2x + 3\).

Step 2: Compare Slopes and Y-Intercepts

The slope of both lines is \(-2\), which means both lines have the same slope, indicating they could either be parallel or identical. However, the first line, \(y = -2x\), has a y-intercept of \(0\), and the second line, \(y = -2x + 3\), has a y-intercept of \(3\). Since the y-intercepts are different, the lines are parallel.

Step 3: Determine Number of Solutions

Since the lines have the same slope but different y-intercepts, they do not intersect at any point. Therefore, the system has no solutions, as parallel lines never meet.

Key Concepts

Parallel Lines

Parallel lines are a foundational concept in geometry and algebra. They run in the same direction, never meeting, similar to railroad tracks. You can always identify parallel lines by their slopes:

• If two lines have identical slopes but different y-intercepts, they are parallel.
• Parallel lines will never cross or touch, meaning there is no point that the two lines share.

Understanding parallel lines is crucial when analyzing systems of equations. In this case, if lines are parallel, it means that they do not intersect anywhere on a graph. Hence, they cannot have common solutions, because there's no point where they meet.

Slope-Intercept Form

The slope-intercept form of an equation is one of the most useful formats for understanding and graphing linear equations. This form is given by:\[ y = mx + b \] where:

• \( m \) is the slope of the line. The slope indicates the steepness or angle of the line, describing how much \( y \) changes with \( x \).
• \( b \) is the y-intercept. The y-intercept is where the line crosses the y-axis.

This form allows easy comparison between different lines:

• Lines with the same slope but different y-intercepts are parallel.
• Lines with different slopes will intersect unless they are vertical or parallel.

In our example, transforming both equations into this form allowed us to quickly identify that they are parallel since both had a slope of \(-2\) but different y-intercepts.

Number of Solutions

The number of solutions in a system of equations tells us about how many points the lines share in common. This can vary depending on the relative positions of the lines:

• If lines intersect at a single point, the system has one solution.
• If lines are parallel, like in our example, there are no solutions, as they never meet.
• If lines are identical, they have infinitely many solutions because every point on one line is also on the other.

Understanding how lines interact in terms of their number of solutions is important for solving systems of equations. In our exercise, because the lines had the same slope but different y-intercepts, they were parallel and thus had no points of intersection. Therefore, there were no solutions to the system.