When solving a system of equations by substitution, what is meant by a useless result $(12=12)$?
A. parallel lines
B. no solution
C. inconsistent system
D. dependent system
Answer (2)
A 'useless result' like 12 = 12 in solving a system of equations by substitution means the variables cancel out, leaving a true but unhelpful statement.
This indicates that the equations are dependent, meaning they represent the same line or plane and have infinitely many solutions.
Parallel lines or an inconsistent system lead to no solution, not a 'useless result'.
Therefore, the correct answer is d . d e p e n d e n t sys t e m .
Explanation
Understanding the Useless Result When solving a system of equations using substitution, a 'useless result' like 12 = 12 means that after substituting and simplifying, the variables cancel out, leaving a true statement that doesn't help us find specific values for the variables. This indicates a special relationship between the equations in the system.
Analyzing the Answer Choices Let's consider what each of the answer choices implies:
Parallel lines: This applies to systems of two linear equations in two variables. Parallel lines have the same slope but different y-intercepts, meaning they never intersect. This leads to no solution , not a useless result like 12 = 12 .
No solution: As mentioned above, this occurs when the equations are inconsistent, such as representing parallel lines.
Inconsistent system: This is another way of saying there's no solution. The equations contradict each other.
Dependent system: This means the equations are essentially the same. One equation is a multiple of the other, or they can be manipulated to be identical. This means they represent the same line (or plane in higher dimensions), and there are infinitely many solutions . When you try to solve by substitution, you end up with an identity like 12 = 12 .
Determining the Correct Answer Since a 'useless result' like 12 = 12 indicates infinitely many solutions and that the equations are dependent, the correct answer is:
d. dependent system
Examples
Imagine you're trying to figure out how much to charge for lemonade at your lemonade stand. You realize that you can either charge $0.50 per cup and make a certain profit, or charge $1.00 per cup and make twice the profit. If you set up a system of equations to model this, you might find that the equations are dependent, meaning they essentially tell you the same thing. This 'useless result' tells you that there are many possible prices and profits that work, as long as the profit scales linearly with the price. You'll need more information (like how much people are willing to pay) to decide on the best price.
A 'useless result' like 12 = 12 in a system of equations means the equations are dependent, indicating they represent the same solution set. This occurs when you substitute and end up with a true but unproductive statement, confirming infinitely many solutions. Therefore, the correct answer is d. dependent system .
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