Problem 75 - Intermediate Algebra

Question

Suppose the graphs of the two linear equations of a system are the same line. What is wrong with this statement? The system has infinitely many solutions. Any ordered pair is a solution of the system.

Step-by-Step Solution

Answer

The error is in claiming any ordered pair is a solution; only those on the line are valid solutions.

Step 1: Identify the error in the statement

The statement says, 'Any ordered pair is a solution of the system.' However, this is incorrect as ordered pairs must satisfy the equation of the line to be considered solutions, not just any ordered pair.

Step 2: Review the characteristics of the statement: "Graphs are the same line"

When the graphs of two linear equations are the same line, it means the equations are equivalent and represent the same set of points on the plane.

Step 3: Define the criteria for "Infinitely many solutions"

Infinitely many solutions occur when two equations are identical and overlap perfectly, representing every point on the shared line.

Step 4: Clarify what subsets of ordered pairs are solutions

Only ordered pairs that satisfy the actual line equation are solutions, not any ordered pair. Thus, the initial statement is incorrect unless those pairs lie on the line.

Key Concepts

Linear Equations

A linear equation is a mathematical statement that represents a straight line when graphed on a coordinate plane. They are typically written in the form \(y = mx + b\), where:

• \(m\) represents the slope of the line, which indicates the line's steepness.
• \(b\) is the y-intercept, where the line crosses the y-axis.

Linear equations can have one, none, or infinitely many solutions, depending on how they relate to each other when forming a system of linear equations. In systems of linear equations, each equation is graphed as a line, and the solution is any point (ordered pair) where the lines meet or overlap. If the lines intersect at one point, there is a unique solution. If the lines are parallel, there are no solutions.

Infinite Solutions

When a system of linear equations has infinite solutions, it means that the two lines represented by the equations are not just similar, but identical. This occurs when:

• Both equations have the same slope \(m\).
• Both equations share the same y-intercept \(b\).
• The lines perfectly overlap, showing they are equivalent.

In this scenario, every point on the line is a solution to both equations, resulting in infinitely many solutions. This commonly happens when one equation is a multiple of another. For example, if one equation is \(y = 2x + 3\) and another is \(2y = 4x + 6\), both equations describe the same line. Therefore, there are countless points (ordered pairs) that solve both equations.

Ordered Pairs

In the context of linear equations, an ordered pair \((x, y)\) represents a specific point on the coordinate plane. It is a solution to a linear equation if it satisfies the equation, meaning when substituted into the equation, both sides equal. For example, if the equation is \(y = 3x + 2\), the ordered pair \((1, 5)\) is a solution because substituting gives \(5 = 3(1) + 2\), which is true.
When dealing with systems of equations that result in infinite solutions, not every ordered pair is a solution, but only those that lie on the line. Thus, it's essential to check if an ordered pair satisfies the equations making up the system. Although multiple pairs might resemble solutions, only those lying precisely on the shared line qualify as solutions. For systems with infinite solutions, consistency and equivalence in representation are key.