Problem 48 - 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 . \(\left\\{\begin{array}{l}2 y=x+2 \\ y+2 x=3\end{array}\right.\)

Step-by-Step Solution

Answer

The lines intersect at a single point; the system has one solution.

Step 1: Understanding the System

We have a system of two linear equations: 1. \( 2y = x + 2 \) 2. \( y + 2x = 3 \). We need to determine if these lines are identical, parallel, or intersecting.

Step 2: Rearrange Equation 1 to Slope-Intercept Form

For the first equation, \( 2y = x + 2 \), solve for \( y \): \[ y = \frac{1}{2}x + 1 \] This is in the slope-intercept form \( y = mx + b \), where \( m = \frac{1}{2} \) and \( b = 1 \).

Step 3: Rearrange Equation 2 to Slope-Intercept Form

For the second equation, \( y + 2x = 3 \), solve for \( y \): \[ y = -2x + 3 \] This is in the slope-intercept form \( y = mx + b \), where \( m = -2 \) and \( b = 3 \).

Step 4: Compare Slopes and Intercepts

The first equation \( y = \frac{1}{2}x + 1 \) has a slope of \( \frac{1}{2} \) and y-intercept of 1. The second equation \( y = -2x + 3 \) has a slope of -2 and y-intercept of 3.

Step 5: Determine Relationship Between Lines

Since the slopes \( \frac{1}{2} eq -2 \), the lines have different slopes and are therefore not parallel or identical. Different slopes mean the lines intersect at exactly one point.

Step 6: Conclusion on Solutions

The lines intersect at a single point, which means the system of equations has exactly one solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations. This form is particularly helpful for understanding the characteristics of a line, such as its slope and y-intercept. A linear equation in slope-intercept form looks like this:

• The standard equation is: \( y = mx + b \)
• "\( m \)" represents the slope of the line, which tells you how steep the line is.
• "\( b \)" indicates the y-intercept, the point where the line crosses the y-axis.

By converting equations into this form, we can easily compare their slopes and y-intercepts. Understanding this form is crucial in solving problems related to the position and interaction of lines.
In our example, we converted the equations to \( y = \frac{1}{2}x + 1 \) and \( y = -2x + 3 \). This enables us to analyze the lines' slopes \( \frac{1}{2} \) and \( -2 \) and intercepts 1 and 3, respectively.

Parallel Lines

Parallel lines are two or more lines that never intersect. They stay the same distance apart over their entire length. For lines to be parallel, their slopes must be identical:

• If two linear equations have the same slope values but different y-intercepts, the lines are parallel and will never meet.

In mathematical terms, if \( m_1 = m_2 \), the lines are parallel. For instance, in a system of equations, you might encounter two lines such as \( y = 2x + 3 \) and \( y = 2x - 4 \). Since both have slopes of 2, they are parallel and do not touch each other anywhere on the plane.
In our original problem, the slopes were \( \frac{1}{2} \) and \(-2\), which are not equal. Thus, the lines are not parallel.

Intersection of Lines

The intersection of lines occurs at the point where two lines meet or cross each other on a graph. If two lines intersect, they have a single solution in the context of a system of linear equations. You can determine if lines will intersect by comparing their slopes:

• If the slopes of two lines are different, the lines will definitely intersect at one point.
• At the intersection point, both equations in the system share the same solution set, i.e., the same \( x \) and \( y \) values.

In the example given, the lines represented by \( 2y = x + 2 \) and \( y + 2x = 3 \) have slopes of \( \frac{1}{2} \) and \(-2\), respectively. Since these slopes are not equal, the lines intersect at a unique point, confirming that the system has exactly one solution.