Solve the system using an augmented matrix.
-x-3y=-17
2x-6y=-26
Can someone please explain what I am supposed to do in this problem?
-1 -3 | -17
2 -6 | -26
add 2x#1 to #2:
-1 -3 | -17
0 -12 | -60
subtract #2 from 4x#1:
-4 0 | -8
0 -12 | -60
Now divide so the diagonal is all 1's:
1 0 | 2
0 1 | 5
x=2, y=5
Sure, I'll explain it to you with a humorous twist!
Alright, so imagine this problem is like a puzzle. And to solve this puzzle, we're going to use a special tool called an augmented matrix. It's like a magic matrix that has been blessed by the math gods!
First, we're going to take the coefficients of x and y from each equation and put them in the matrix. But wait, hold on tight! We can't forget about that sneaky little equal sign! We're going to add an extra column in the matrix to keep track of it.
Here's what the matrix will look like:
_ _ _ | _
-1 -3 -17
2 -6 -26
Now that our augmented matrix is all set up, we can start the fun part - row operations! The goal is to transform the matrix into something called "reduced row-echelon form." It sounds fancy, but it's just a fancy way of saying "solving the system." We're basically going to play a game of matrix dress-up!
Start by swapping the rows around if needed, because matrix fashion is all about looking fabulous! But beware, don't overdo it! We don't want a messy matrix.
Once you've got the rows in their proper places, it's time to use some clever multiplication and addition to eliminate those pesky coefficients. You're like a magician performing matrix wizardry!
Multiply a row by a constant? Sure, why not! Just be careful not to turn the matrix into a math maze. We don't want to get lost!
Add one row to another row? Absolutely! It's like giving them a high-five, but with numbers. Don't worry, they won't get hurt. They're tough!
Keep performing row operations until you have a beautifully dressed-up matrix, where the left side has a diagonal of ones and the right side has the solutions to x and y. It's like having a perfect outfit for x and y to wear to the math party!
Once you've achieved this elegant look, you can read off the solutions for x and y from the right side of the matrix.
So, keep rocking those row operations, and you'll soon find the solutions that will make this system of equations say, "Wow, you really cracked the code!"
Good luck, my matrix-solving friend!
To solve the system of equations using an augmented matrix, you will follow these steps:
1. Write down the coefficients and constants of the variables in matrix form. The first row represents the coefficients and constants of the first equation, and the second row represents the coefficients and constants of the second equation.
-x -3y = -17
2x -6y = -26
In matrix form, it becomes:
| -1 -3 | -17 |
| 2 -6 | -26 |
2. Perform elementary row operations to transform the augmented matrix into row-echelon form or reduced row-echelon form. The row-echelon form is a matrix whose leading entry (first non-zero element) of each row is a 1 and is to the right of the leading entry of the row above it.
3. Use back substitution or Gaussian elimination to find the values of x and y from the row-echelon form or reduced row-echelon form.
Let's go through the steps to solve the given system of equations.
Is there anything else you would like to ask?
To solve the system of equations using an augmented matrix, you need to follow these steps:
1. Write the system of equations in matrix form. This involves rewriting the equations with all the coefficients, variables, and constants on one side of the equation, and setting up a matrix with the coefficients of the variables and the constants as the last column. From the given system of equations:
-x - 3y = -17
2x - 6y = -26
The matrix form will be:
| -1 -3 | -17 |
| 2 -6 | -26 |
2. Perform row operations on the matrix to simplify and solve the system. The goal is to transform the matrix into reduced row-echelon form, where the variables are easily solvable. The typical row operations include multiplying or dividing a row by a non-zero constant, adding or subtracting multiples of one row from another row, and interchanging rows.
Let's start by making the first entry in the first column (the leading coefficient) equal to 1. Multiply the first row by -2 and add it to the second row:
| -1 -3 | -17 |
| 0 0 | 8 |
3. Next, we'll multiply the first row by -1 and add it to the second row to eliminate the first variable from the second row:
| -1 -3 | -17 |
| 0 0 | 8 |
This matrix is now in reduced echelon form. The second row represents the equation 0x + 0y = 8, which simplifies to 0 = 8. This equation is inconsistent and does not have a solution.
Therefore, the system of equations is inconsistent, and there is no solution.
I hope this explanation helps you understand how to solve the given system of equations using an augmented matrix.