Problem 62 - Introductory Algebra for College Students

Question

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. Each equation in a system of linear equations has infinite many ordered-pair solutions.

Step-by-Step Solution

Answer

The statement does not make sense. Normally, each equation in a system of linear equations cannot have infinitely many ordered-pair solutions. However, under specific circumstances where all equations coincide and represent the same line, the statement could be true.

Step 1: Understanding of an Individual Linear Equation

Considering an equation in a system of linear equations, an equation of a linear equation, assuming two variables, represents a line. A line has infinitely many points, hence infinitely many ordered pairs, but these aren't all solutions to the equation. Each point corresponds to an ordered pair (x, y). A solution of a linear equation is an ordered pair that satisfies the equation, making the statement false.

Step 2: Understanding of a System of Linear Equations

Considering a set of linear equations (a system), it can have zero, one or infinitely many solutions. This depends on whether the lines represented by the equations are parallel (no solution), intersect at a point (one solution) or coincide (infinitely many solutions). The latter case implies that all equations represent the same line, thus have the same ordered pairs solutions, making it possible but not the common case. The provided statement is wrong under normal circumstances however could be true in certain unique scenarios.

Key Concepts

Linear Equation

A linear equation in mathematics refers to an equation that represents a straight line in a coordinate system. It is typically expressed in the form \(y = mx + b\), where \(m\) stands for the slope of the line and \(b\) is the y-intercept. This means:

• The slope \(m\) describes how steep the line is.
• The y-intercept \(b\) is where the line crosses the y-axis.

Every linear equation forms an infinite line made up of points. These points translate into *ordered pairs* \((x, y)\) that satisfy the equation. By plugging any value for \(x\) into the equation, you can calculate the corresponding value of \(y\), forming a complete solution pair. However, remember not all points on a line will be solutions to a specific linear equation problem, only those fulfilling the equation's criteria.

Ordered Pair Solutions

In mathematics, ordered pairs are written as \((x, y)\). This tuple describes the coordinates of a point on a two-dimensional graph.

• When addressing solutions for linear equations, these pairs represent solutions where both \(x\) and \(y\) satisfy the equation.
• For example, in the linear equation \(y = 2x + 1\), the ordered pair \((2, 5)\) is a solution because if you substitute \(x = 2\), you get \(y = 5\).

Ordered pairs are crucial in graphing as they map the entire curve of the line. In a system of linear equations, finding common solutions refers to identifying the points where the graphs of the equations cross or overlap, which are also valid solutions for all the equations in that system.

Infinite Solutions

A system of linear equations may sometimes have infinite solutions. This occurs when the equations describe the same line.

• For each point on the line, both equations are satisfied, meaning they have zero discrepancy.
• In essence, every point on the line \((x, y)\) satisfies all the equations in this system.

This typically happens when two linear equations are scalar multiples of each other, effectively describing the same geometric plane. For example, if one equation is \(2x + 3y = 6\) and the other is simply a multiple \(4x + 6y = 12\), they are different expressions of the same line, thus having infinite solutions.

Intersecting Lines

In a system of linear equations, intersecting lines tell us important information about solutions.

• When two lines intersect at a single point, it means that the system of equations has one unique solution represented by that intersection.
• The coordinates of the intersection point (an ordered pair) satisfy both equations simultaneously, providing the solution to the system.

Apart from unique solutions at intersections, remember that parallel lines never intersect, resulting in no solutions. Understanding intersecting lines helps identify how many solutions a system has, and it highlights the concept of dependency and independency among equations.