91Ó°ÊÓ

The substitution method is a technique used to solve systems of equations where one equation is solved for one of the variables, and then this expression is substituted into the other equation. This allows for solving one of the variables directly. In the exercise given, we have two equations:

• 5x - 2y = 19
• y = \( \frac{1 - 3x}{4} \)

The second equation is already solved for \( y \), making it a perfect candidate for substitution. By inserting the expression for \( y \) from the second equation into the first equation, we replace \( y \) with \( \frac{1 - 3x}{4} \). Then, the substituted equation becomes \( 5x - 2 \left( \frac{1 - 3x}{4} \right) = 19 \). This substitution process converts the problem into a single-variable equation, which is often easier to solve than dealing with a system of two variables.

Linear equations form the backbone of solving systems like this. These are equations in which each term is either a constant or the product of a constant and a variable. A simple linear equation takes the form \( ax + by = c \). For solutions, linear equations often appear in pairs or sets that require solving systems. Each equation in a system represents a line on a graph, and solving the system is essentially finding the point where the lines intersect.

In our original example, both equations establish a system of linear equations:

• The first equation is \( 5x - 2y = 19 \)
• The second equation when fully simplified would relate to the line formed by \( y = \frac{1 - 3x}{4} \)

These lines cross at the point \((3, -2)\) in the solution rendering them consistent, unlike some other systems which we will discuss.

Inconsistent systems and dependent systems are special occasions in the study of linear equations. An inconsistent system occurs when the two lines represented by the equations in a system do not intersect at any point; this means there is no solution that satisfies both equations simultaneously. Typically, these lines will be parallel and have the same slope but different y-intercepts, leading them never to meet.

A dependent system, on the other hand, arises when the equations describe the same line. Thus, there are infinitely many solutions since every point on the line is a solution to both equations. In such cases, one equation is a multiple of the other. When solving, you’ll notice they reduce to an identity, like \( 0 = 0 \), indicating their equivalence.

In our exercise, the two equations form a consistent system - meaning they meet at precisely one point \((3, -2)\) and resolve to a single solution. They do not exhibit characteristics of being either inconsistent or dependent.