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1. Consider the following system of equations. 3x_1 - 2x_2 = 5 \\ px_1 + 4x_2 = c (a) For which...

Question:

1. Consider the following system of equations.

{eq}3x_1 - 2x_2 = 5 \\ px_1 + 4x_2 = c {/eq}

(a) For which values of p and c is the above system consistent? Explain.

(b) For which values of p and c is the above system inconsistent? Explain.

2. Mr. Jones claims that if a system of equations has the same number of variables as the number of equations, the system is ALWAYS consistent. Is he correct? Explain.

Consistent System of Equations

This problem is based on the concepts of the consistency of the equations and the conditions for the equations to be consistent. We are going to apply the conditions for the consistency of the equations and determine the values of the variables.

Answer and Explanation:


Part 1:

The set of equations that we have are:

{eq}3x_1 - 2x_2 = 5 \\ px_1 + 4x_2 = c {/eq}

Part A:

We need to calculate the values of {eq}\displaystyle p {/eq} and {eq}\displaystyle c {/eq}.

Assuming that the pair of the equation is consistent and is dependent. We should have,

{eq}\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} {/eq}

Thus,

{eq}\displaystyle \frac{3}{p}=\frac{-2}{4}=\frac{5}{c} {/eq}

So,

{eq}\displaystyle \frac{3}{p}=\frac{-2}{4} {/eq}

{eq}\displaystyle p=-6 {/eq}

{eq}\displaystyle \frac{-2}{4}=\frac{5}{c} {/eq}

{eq}\displaystyle c=-10 {/eq}

Thus, the required values of {eq}\displaystyle p {/eq} and {eq}\displaystyle c {/eq}:

{eq}\displaystyle \boxed{p=-6 \text{ and }c=-10} {/eq}


Part B:

For the system to be inconsistent, we have,

{eq}\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2} {/eq}

So,

{eq}\displaystyle \frac{3}{p}=\frac{-2}{4}\neq \frac{5}{c} {/eq}

Thus,

{eq}\displaystyle \boxed{\displaystyle p=-6} {/eq}

And,

{eq}\displaystyle \frac{-2}{4}=\frac{5}{c} {/eq}

{eq}\displaystyle \boxed{\displaystyle c \neq -10} {/eq}

So, the set of the values of {eq}\displaystyle c {/eq} for which the equation is inconsistent is,

{eq}\displaystyle \boxed{\text {It could be any value of }c \text{ except 10}.} {/eq}


Part 2:

We have to explain if the claim is correct or not.

The claim states:

{eq}\displaystyle \text{Mr. Jones claims that if a system of equations has the same number of variables as the number of equations, the system is ALWAYS consistent.} {/eq}

  • If the system of equations have the same number of variables as the number of equations IT IS NOT necessarily consistent because for the system of equations to be consistent the coefficients of the variables must follow either {eq}\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} {/eq} OR {eq}\displaystyle \frac{a_1}{a_2}\neq \frac{b_1}{b_2} {/eq} (for a two variable system).

Learn more about this topic:

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Consistent System of Equations: Definition & Examples

from High School Algebra II: Homework Help Resource

Chapter 8 / Lesson 8
19K

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