Problem 45 - Introductory Algebra for College Students

Question

When using the substitution method, how can you tell if a system of linear equations has no solution?

Step-by-Step Solution

Answer

A system of linear equations has no solution using the substitution method when the substitution leads to an impossible or untrue statement, such as 0 equals a non-zero number.

Step 1: Understand the Substitution Method

In the substitution method, the goal is to express one variable from one equation in terms of the other variable(s), and then substitute this expression into the other equation(s). As a result, one should get an equation which involves only one variable and can be solved directly.

Step 2: Apply the Substitution

Substitute the expression obtained in step one into the other equation(s). This should provide an equation involving only one variable, which can then be solved.

Step 3: Determine the Absence of Solution

In cases where the equations have no solution, you would obtain a contradiction during the substitution process. For instance, you might obtain a statement like \(0=3\), which is clearly untrue.

Step 4: Example

Consider two equations, \(x+2y=3\) and \(2x+4y=7\). Here, substituting \(x = 3 - 2y\) into the second equation results in \(2(3-2y) + 4y = 7\), which simplifies to \(6-4y +4y=7\) or \(6=7\), a contradiction, thus indicating that the system has no solution.