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Chapter 7: Problem 73

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

Short Answer

Expert verified
A system of linear equations has no solution if, after using addition/substitution, a false statement results (like 0=5). This corresponds to parallel lines on a graph. The relationship between the two graphs is crucial in determining the solution type.

Step by step solution

01

Understand the concept of system of linear equations

A system of linear equations is a set of two or more linear equations that all contain the same set of variables. A solution to a system of linear equations is a value (or set of values) for the variables that satisfy all the equations simultaneously.
02

Use of addition or substitution method

When using the addition method, the two equations are added together to eliminate one variable, allowing you to solve for the other variable. In the substitution method, you solve one equation for one variable and then substitute this term into the other equation. In either case, if you get a false statement (like 0 = 5), this indicates that the system has no solution.
03

Graphical interpretation

The graphical interpretation of system of linear equations is that each equation represents a line in a coordinate plane. When the system has no solution, the lines represented by the equations are parallel. When there is a single solution, the two lines intersect at a single point. If there are infinitely many solutions, the lines coincide or are overlapping.

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Most popular questions from this chapter

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing \(3 x-4 y<12\), it's not necessary for me to graph the linear equation \(3 x-4 y=12\) because the inequality contains a \(<\) symbol, in which equality is not included.

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