91Ó°ÊÓ

A system of equations is a collection of two or more equations with the same set of variables. In our exercise, we have two equations with variables \(p\) and \(q\): \[\begin{align*}3p - q &= 7, \-3p + q &= \frac{3}{4}.\end{align*}\] The goal is to find values for \(p\) and \(q\) that satisfy both equations simultaneously. These systems can have one solution, infinitely many solutions, or no solution at all.

• One Solution: This means the lines intersect at a single point.
• Infinitely Many Solutions: This occurs when the lines are the same and overlap completely.
• No Solution: This indicates the lines are parallel and never meet.

In our case, the system is inconsistent, which we'll delve into more as we examine the characteristics of linear equations and parallel lines.

Linear equations are the most basic type of algebraic equations. They form straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables \(x\) and \(y\) is \(ax + by = c\). In our system, the linear equations are: \[\begin{align*}3p - q &= 7, \-3p + q &= \frac{3}{4}.\end{align*}\] Linear equations can be solved using various methods such as substitution, elimination, or graphing. Each equation in a system represents a line, and the solution to the system is the intersection point of these lines, if it exists. Here, we attempted to eliminate \(q\) by adding the equations: \[(3p - q) + (-3p + q) = 7 + \frac{3}{4}.\] This resulted in an untrue statement \(0 = \frac{31}{4}\), confirming no intersection point exists.

Parallel lines are two or more lines in a plane that never intersect. They always stay the same distance apart, no matter how far they are extended. For two lines to be parallel, they must have the same slope but different y-intercepts. In our exercise, we had: \[\begin{align*}3p - q &= 7, \-3p + q &= \frac{3}{4}.\end{align*}\] Upon manipulation, we found \(0 = \frac{31}{4}\), indicating that the equations represent parallel lines. This is because simplifying them leads to equivalent slopes, but since there's no common solution, it confirms that they are parallel. Parallel lines are a key concept in understanding inconsistent systems, as they visually represent the idea of non-intersecting lines, thus reflecting the property of having no solutions.