Problem 91 - College Algebra Essentials

Question

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two equations?

Step-by-Step Solution

Answer

A system of linear equations has no solution if, by applying the addition or substitution method, a false statement like 0=5 is obtained. This means that the two lines represented by these equations are parallel.

Step 1: Understanding Addition and Substitution Method

A system of linear equations can be solved using either the addition or substitution method. In the substitution method, one of the equations is solved for one variable (say, x) which is then substituted in the other equation. In the addition method, the equations are added or subtracted from each other such that one of the variables cancels out, giving an equation in only one variable (x or y).

Step 2: Checking Solutions

After using the addition or substitution method, the results can be classified into three categories. Firstly, if manipulating the equations results in a true statement (like 0=0), then it means the system of equations has infinitely many solutions. Secondly, if it results in a false statement (like 0=5), it means the system of equations has no solution. Lastly, if it results in a singular equation (like x=3), the system of equations has a unique solution.

Step 3: Graphical Interpretation

In graphical terms, if the two lines represented by the system of linear equations coincide, then the system has infinitely many solutions. If the two lines are parallel, then the system has no solution as parallel lines do not intersect. If two lines intersect at a single point, then the system has a unique solution, represented by the coordinates of that point.