Problem 31 - Algebra Form and Function

Question

Without solving the equations, decide how many solutions the system has. $$ \left\\{\begin{array}{r} 3 y=2-x \\ x=2+3 y \end{array}\right. $$

Step-by-Step Solution

Answer

Equations: 1) 3y = 2 - x 2) x = 2 + 3y Answer: The system of equations has exactly one solution.

Step 1: Identify the slopes of the lines

We will rewrite each equation in the form y = mx + b, where m is the slope, x is the independent variable, and b is the y-intercept. For the first equation: $$ 3y = 2 - x \\ y = -\frac{1}{3}x + \frac{2}{3} $$ The slope of the first line is \(-\frac{1}{3}\). For the second equation: $$ x = 2 + 3y \\ y = \frac{1}{3}x - \frac{2}{3} $$ The slope of the second line is \(\frac{1}{3}\).

Step 2: Compare the slopes of the lines

Since the slopes are different (first line has a slope of \(-\frac{1}{3}\) and the second line has a slope of \(\frac{1}{3}\)), the lines are not parallel and will intersect at some point.

Step 3: Determine the number of solutions

As the lines have different slopes and are not parallel, they will intersect at exactly one point. Therefore, this system of equations has exactly one solution.

Key Concepts

Slopes of Lines

The slope of a line is a measure of its steepness, indicating the vertical change for a given horizontal change between two points on the line. In mathematical terms, this is often denoted as \( m \) in the line equation form \( y = mx + b \), where \( b \) is the y-intercept. Understanding the slope helps in predicting how the line behaves as you move along it. A positive slope (e.g., across from bottom left to top right) indicates that as one moves to the right, the line rises. Conversely, a negative slope means the line falls as you move to the right. For example, if a line has a slope of \(-\frac{1}{3}\), you would know that for each step to the right on the x-axis, the line moves one-third of a step down on the y-axis. Meanwhile, a positive slope of \(\frac{1}{3}\) signifies movement upward one-third of a step per rightward step. This concept is essential when analyzing systems of equations to understand how the lines relate to one another.

Intersection of Lines

The intersection of lines occurs at a point where the graphs of those lines meet. In the context of linear equations, determining this point of intersection is crucial for solving systems of equations. This indicates the solution to the system and where both conditions set by the equations are satisfied simultaneously.If two lines have different slopes, they will intersect at one point because they are not parallel. If the lines are parallel (identical slopes), they do not meet, hence no intersection unless they are, in fact, the same line repeated. For our system of equations:

• The first equation has a slope of \(-\frac{1}{3}\)
• The second equation has a slope of \(\frac{1}{3}\)

These different slopes ensure they will intersect at exactly one unique point, meaning the system has one solution.

Linear Equations

Linear equations represent straight lines on a graph and are written in various formats, such as \( y = mx + b \) or \( ax + by = c \). They consist of variables (usually \( x \) and \( y \)) and constants, and they model a range of practical situations due to their predictable nature. The beauty of linear equations lies in their simplicity and their ability to often solve problems in straightforward, predictable ways. By translating real-world problems into linear equations, one can uncover relationships and solutions that are not immediately obvious. In systems of linear equations involving two or more linear equations, each equation represents a line, and solving the system typically involves finding common solutions or points of intersection. The systems can be categorized as:

• Unique solution: Where lines intersect at a single point.
• No solution: Where lines are parallel and do not meet.
• Infinite solutions: Where lines coincide completely, meaning they are the same line.

Understanding these outcomes helps in predicting the nature of solutions for problems modeled by linear equations.