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Chapter 7: Problem 40

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{l}9 x-3 y=12 \\ y=3 x-4\end{array}\right.\)

Short Answer

Expert verified
The system of equations has an infinite number of solutions, expressed in set notation as \{(x, y) | y = 3x - 4, x in R\}, where R represents the set of real numbers.

Step by step solution

01

Substitute for y in the First Equation

Substitute \(y = 3x - 4\) from the second equation into the first equation, resulting in \(9x - 3(3x - 4) = 12\).
02

Simplify the Equation

Simplify this equation, which results in an equation \(9x - 9x + 12 = 12\), further simplifying to \(12 = 12\).
03

Possible Solutions

This results in a true statement, indicating that the system of equations has an infinite number of solutions. Which means the values of x and y such that \(y = 3x - 4\) are the solutions for the system.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{r}x+y<4 \\ 4 x-2 y<6\end{array}\right.\)

Graph each linear inequality. \(y<-\frac{1}{4} x\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \leq 5 \\ y>-3\end{array}\right.\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x-y \leq 1 \\ x \geq 2\end{array}\right.\)

Without graphing, Determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}3 x+y \leq 9 \\ 3 x+y \geq 9\end{array}\right.\)
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