Solve the system by using Gaussian elimination or Gauss-Jordan elimination.
[tex]
\begin{aligned}
-4 x+11 y & =58 \\
x-3 y & =-12
\end{aligned}
[/tex]
The solution set is { [ ] , [ ] )}
Answer (2)
Write the augmented matrix for the system.
Perform row operations to get the matrix in reduced row-echelon form.
Read the solution directly from the reduced row-echelon form.
The solution set is ( − 42 , − 10 ) .
Explanation
Understanding the Problem We are given a system of two linear equations in two variables, x and y . Our goal is to solve this system using Gaussian elimination or Gauss-Jordan elimination.
Stating the Equations The given equations are:
− 4 x + 11 y = 58 x − 3 y = − 12
Writing the Augmented Matrix First, we write the augmented matrix for the system: [ − 4 1 11 − 3 58 − 12 ]
Swapping Rows Next, we swap row 1 and row 2 to get a 1 in the top left corner: [ 1 − 4 − 3 11 − 12 58 ]
Eliminating x in the Second Row Now, we add 4 times row 1 to row 2 to eliminate the x term in the second equation: [ 1 − 4 + 4 ( 1 ) − 3 11 + 4 ( − 3 ) − 12 58 + 4 ( − 12 ) ] = [ 1 0 − 3 − 1 − 12 10 ]
Isolating y in the Second Row Multiply row 2 by -1 to get a 1 in the second row, second column: [ 1 0 − 3 1 − 12 − 10 ]
Eliminating y in the First Row Finally, add 3 times row 2 to row 1 to eliminate the y term in the first equation: [ 1 + 3 ( 0 ) 0 − 3 + 3 ( 1 ) 1 − 12 + 3 ( − 10 ) − 10 ] = [ 1 0 0 1 − 42 − 10 ]
Stating the Solution From the reduced row-echelon form, we can directly read the solution: x = − 42 and y = − 10 . Therefore, the solution set is ( − 42 , − 10 ) .
Examples
Systems of linear equations are used in various fields such as engineering, economics, and computer science. For instance, in electrical engineering, they can be used to analyze circuits. In economics, they can model supply and demand curves to find equilibrium prices. In computer graphics, they are used to perform transformations on objects.
To solve the system of equations using Gaussian elimination, we derived the reduced row-echelon form and found that the solution is ( − 42 , − 10 ) . We transformed the augmented matrix step-by-step, eliminating variables to find the values of x and y directly. The final answer indicates that x equals -42 and y equals -10.
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