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Chapter 4: Problem 60

What are dependent equations? Provide an example with your description.

Short Answer

Expert verified
Dependent equations are those in a system that do not add any new information to the system, i.e., they can be derived from the other equations in the system. For example, in the system of equations containing '2x + 3y = 6' and '4x + 6y = 12', the second equation is just the first equation multiplied by 2, so these equations are dependent.

Step by step solution

01

Understanding Dependent Equations

Dependent equations are equations in a system that do not add any new information, and they can be derived from other equations in the system. This usually means that they have the same solution set as other equations in the system. The name dependent comes from the fact that these equations 'depend' on other equations in the system. For instance, if in a system of equations, the second equation is just a multiple of the first, those equations are dependent because the second can be obtained from the first by multiplication.
02

Example of Dependent Equations

Here is a simple example to illustrate dependent equations:Consider the following two equations:1) \(2x + 3y = 6\)2) \(4x + 6y = 12\)In this system, the second equation can be directly derived from the first equation by multiplying all the terms of the first equation by \(2\). Thus the second equation doesn't add any new information to the solution set of the system, and is, therefore, dependent on the first equation.

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Most popular questions from this chapter

In Exercises \(80-83,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(A x+2 y=2\) and \(2 x+B y=10\) have graphs that intersect at \((2,-2),\) then \(A=-3\) and \(B=3\)

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{x}{2}=\frac{y+8}{3} \\ \frac{x+2}{2}=\frac{y+11}{3} \end{array}\right.$$

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} y=2 x+4 \\ y=2 x-1 \end{array}\right.$$

Solve: \(29,700+150 x=5000+1100 x\)

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} 2(x+y)=4 x+1 \\ 3(x-y)=x+y-3 \end{array}\right.$$
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