Problem

Use the echelon method to solve the following system of two equations in two unknowns. Chèck your answer. \[ \begin{aligned} 4 x-3 y & =-5 \\ -8 x+6 y & =10 \end{aligned} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is (Type an ordered pair.) B. There are infinitely many solutions. The solution is $(\square, y)$, where $y$ is any real number. C. There is no solution.

Solution

Step 1 :The given system of equations is: \[\begin{aligned} 4 x-3 y & =-5 \\ -8 x+6 y & =10 \end{aligned}\]

Step 2 :We can see that the second equation is just the first equation multiplied by -2. This means that the two equations are dependent and represent the same line.

Step 3 :Therefore, there are infinitely many solutions to this system of equations. However, we can still find the general form of the solutions.

Step 4 :The solution to the system of equations is given by the expression \(x = \frac{3y}{4} - \frac{5}{4}\). This means that for any real number \(y\), we can find a corresponding \(x\) that satisfies both equations.

Step 5 :\(\boxed{\text{Final Answer: There are infinitely many solutions. The solution is }\left(\frac{3y}{4} - \frac{5}{4}, y\right)\text{, where } y \text{ is any real number.}}\)

From Solvely APP
Source: https://solvelyapp.com/problems/38065/

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