Problem 91 - Algebra and Trigonometry

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

Using addition or substitution methods, if the system gives an inconsistent equation (like 0=5), it doesn't have a solution. Graphically, this implies that the two equations yield parallel lines that never intersect. This scenario occurs if coefficients of \(x\) and \(y\) in both equations are proportional while the constants are not.

Step 1: Identifying System Status

To determine whether a system of linear equations has a solution, solve it using either the addition or substitution methods. If the system results in an equation that doesn’t make sense, like 0=5 for instance, it is an inconsistent system and has no solution.

Step 2: Recognizing Parallel Lines

To understand the connection between the system solution and graphical representation, it's necessary to recognize that a system of linear equations represents two lines on a graph. If these lines are parallel (i.e., the equations are proportional), they do not intersect at any point. This means the system doesn't have a solution since there's no point that belongs to both lines.

Step 3: Determining Coefficient Proportions

Coefficients of the two equations in the system can help in assessing whether the lines are parallel. If the ratios for the coefficients of \(x\) and \(y\) between the two equations are equal, while for the constant term it does not hold, it’s a sign that the system has no solution. These coefficients determine the slope of the lines - if the slopes are equal, and the y-intercepts are different, the lines are parallel.