Problem 34 - Intermediate Algebra

Question

Is it possible for a system of two linear equations in two variables to have exactly two solutions? Defend your answer.

Step-by-Step Solution

Answer

No, a system of two linear equations cannot have exactly two solutions.

Step 1: Understanding Linear Equations

A linear equation in two variables is an equation of the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. A system of two linear equations involves two such equations, each for the same variables.

Step 2: Graphing the Equations

When we graph a linear equation in two variables, it results in a straight line. Therefore, a system of two linear equations will produce two lines on the coordinate plane.

Step 3: Analyzing Line Intersections

There are three possible cases for the lines: they intersect at one point, are parallel (and hence never intersect), or are the same line (coinciding).

Step 4: Evaluating Intersections for Two Solutions

If the lines intersect at exactly one point, the system has one solution. If they are parallel, there are no solutions. If they coincide completely, there are infinitely many solutions. Thus, they cannot intersect at exactly two points.

Step 5: Conclusion

A system of two linear equations cannot have exactly two solutions because lines either intersect at one point, do not intersect at all, or are identical.

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra and are widely used in various mathematical applications. They are equations that form straight lines when plotted on a graph. The general form of a linear equation in two variables is given by \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

• \( a \) and \( b \) represent coefficients of the variables \( x \) and \( y \) respectively.
• The variable \( c \) is the constant term or the y-intercept when the equation is rearranged in the slope-intercept form \( y = mx + c \).

Linear equations are characterized by their degree, which is one, since their highest power of the variable is one. This simplicity makes them predictable and easy to work with in solving various mathematical problems.

Solutions to Equations

Solutions to equations refer to the set of values that satisfy the equation. In the context of a system of linear equations, solutions represent points where all equations in the system intersect.

• A solution to a single linear equation can be any of the infinite points along its line.
• For a system of two linear equations, solutions are the points of intersection of the lines.

Based on how these lines interact, we can determine the number of solutions: - **One Solution**: The lines intersect at exactly one point. This scenario is possible if the lines have different slopes. - **No Solution**: The lines are parallel, meaning they never meet. This occurs when the lines have the same slope but different y-intercepts. - **Infinite Solutions**: The lines coincide, meaning they are the same line and overlap completely. This context helps us understand why a system cannot have exactly two solutions since there are no conditions for lines to intersect twice on a plane.

Graphing Linear Equations

Graphing linear equations provides a visual method for analyzing solutions by plotting their graphs on a coordinate plane. Each linear equation, when graphed, results in a straight line.

• The slope of the line \( m \) is determined by the coefficients of the variables \( x \) and \( y \).
• The y-intercept \( c \) is where the line crosses the y-axis, providing a starting point for plotting the graph.

Visualizing linear equations through graphs can simplify understanding of complex solutions and intersections. When graphing a system of linear equations, it's essential to look at the position and slope of each line to determine their interactions. - **Intersection**: Where two graphs cross, indicating a potential solution.- **Parallel**: When lines never meet, illustrating no common solution for the system.- **Overlapping**: Lines perfectly align, suggesting an infinite number of solutions for the system. Graphing makes it easier to see these interactions clearly, highlighting why a system of two linear equations cannot intersect at exactly two points. This process emphasizes the fundamental rule about the behavior of lines on a plane.