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Chapter 5: Problem 29

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{l} 3 x-4 y=6 \\ 9 x=12 y+4 \end{array} $$

Short Answer

Expert verified
The system is inconsistent.

Step by step solution

01

- Write the system of equations

Given the system of equations: \[ \begin{array}{l} 3x - 4y = 6 \ 9x = 12y + 4 \ \end{array} \]
02

- Simplify the second equation

Rewrite the second equation in standard form. Start by moving all terms to one side: \[ 9x - 12y = 4 \]
03

- Compare the equations

Divide the second equation by 3 to simplify: \[ \frac{9x - 12y}{3} = \frac{4}{3} \]This simplifies to: \[ 3x - 4y = \frac{4}{3} \]
04

- Compare coefficients and constants

Compare the simplified second equation \[ 3x - 4y = \frac{4}{3} \] with the first equation \[ 3x - 4y = 6 \].Notice that the coefficients of x and y are the same, but the constants on the right side are different.
05

- Determine if the system is consistent or inconsistent

Since the two equations have the same coefficients for x and y but different constants, there is no solution that satisfies both equations simultaneously. Therefore, the system is inconsistent.

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Key Concepts

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Most popular questions from this chapter

Solve the system using any method. $$ \begin{array}{l} y=\frac{2}{3} x-1 \\ y=\frac{1}{6} x+2 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} 4(x-2)=6 y+3 \\ \frac{1}{4} x-\frac{3}{8} y=-\frac{1}{2} \end{array} $$

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{l} -4 x-8 y=2 \\ 2 x=8-4 y \end{array} $$

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{aligned} 2 x-y &=8 \\ x-\frac{1}{2} y &=4 \end{aligned} $$

Solve the system using any method. $$ \begin{array}{l} \frac{x-2}{8}+\frac{y+1}{2}=-6 \\ \frac{x-2}{2}-\frac{y+1}{4}=12 \end{array} $$
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