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Chapter 7: Problem 41

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. \(\left\\{\begin{array}{l}3 x-2 y=-5 \\ 4 x+y=8\end{array}\right.\)

Short Answer

Expert verified
The solution to the given system of equations is \(x = 1\) and \(y = 4\).

Step by step solution

01

Identifying the Equations

Firstly, you need to identify the equations from the system. The system provided consists of the following equations:Equation (1): \(3x - 2y = -5\)Equation (2): \(4x + y = 8\)
02

Solve one of the equations for one variable

Solve the second equation for y: \(y = 8 - 4x\) This will be useful when substituting y in equation 1.
03

Substitute Value of y in Equation 1

Replace y in Equation 1 with the expression obtained in step 2, which gives us:\(3x - 2(8 - 4x) = -5\), which simplifies to: \(11x - 16 = -5\), thus \(x = 1\)
04

Substitute The Value of x Into Equation 2

Next, substitute the value of x into equation 2 to get a value for y.This gives: \(4*1 + y = 8\), resulting in \(y = 4\)
05

Check the Solution

Finally, check if these values, \(x = 1\) and \(y = 4\), are solutions to both original equations. When these values are input into the first equation, it becomes: \(3*1 - 2*4 = -5\), which is true; and when input into the second equation, it becomes: \(4*1 + 4 = 8\), which is also true. Thus, these values are indeed the solutions to the system and the system has a unique solution.

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Most popular questions from this chapter

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 4 \\ \hline 1 & 1 \\ \hline 2 & 0 \\ \hline 3 & 1 \\ \hline 4 & 4 \\ \hline \end{array} $$

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\).

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 2 \\ y \leq 3\end{array}\right.\)

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

Graph each linear inequality. \(x-y \leq 1\)
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