Write the system of equations as an augmented matrix. Then solve for x and y.
5x-4y=12
-5x+3y=9
+5 -4 12 divide by 5 to get 1
-5 +3 +9
+1 -4/5 12/5 multiply by 5 and add
-5 +3.0 +9
+1 -4/5 12/5
+0 -1.0 21 multiply by -1
+1 -4/5 12/5
+0 +1.0 -21 so y = -21
now multiply second eqn by 4/5 and add to first
+1 -4/5 12/5
+0 +4/5 -84/5
+1 +0 -72/5
+0 +1 -21
x = -72/5, y = -21
To write the system of equations as an augmented matrix, we can simply arrange the coefficients of x and y as well as the constants on the right-hand side of the equations.
The augmented matrix can be written as:
[5 -4 | 12]
[-5 3 | 9]
To solve for x and y, we can use row operations to transform the augmented matrix into row-echelon form or reduced row-echelon form. Let's solve it using the Gaussian elimination method:
First, let's multiply the first equation by -1 to eliminate x:
-1 * [5 -4 | 12] = [-5 4 | -12]
Now, add the second equation to the first equation:
[-5 4 | -12]
+ [-5 3 | 9]
=
[0 7 | -3]
Now, divide the second row by 7 to simplify the equation:
[0 1 | -3/7]
This means that y = -3/7.
Now, substitute y = -3/7 into the first equation:
5x - 4(-3/7) = 12
5x + 12/7 = 12
Multiply the equation by 7 to get rid of the fraction:
35x + 12 = 84
35x = 84 - 12
35x = 72
x = 72/35
Therefore, the solution to the system of equations is x = 72/35 and y = -3/7.
To write the system of equations as an augmented matrix, we will transfer the coefficients and constant terms to a matrix form.
The system of equations is:
5x - 4y = 12
-5x + 3y = 9
The coefficients of x and y, as well as the constant terms, will go into the matrix.
The augmented matrix is formed by arranging the coefficients of x and y, and the constant terms, in a matrix with vertical bars representing the equal sign.
So, the augmented matrix will look like this:
[5 -4 | 12]
[-5 3 | 9]
To solve for x and y using the augmented matrix, we will perform row operations to transform the matrix into row-echelon form or reduced row-echelon form.
Let's now perform the row operations to solve the system of equations.
**Step-by-step Solution:**
1. Perform the required row operations to get a leading coefficient of 1 in the first row.
- Divide the first row by 5: R1/5 -> R1
- The augmented matrix becomes:
[1 -4/5 | 12/5]
[-5 3 | 9]
2. Perform the required row operations to eliminate the coefficient of x in the second row.
- Multiply the first row by 5 and add it to the second row: 5*R1 + R2 -> R2
- The augmented matrix becomes:
[1 -4/5 | 12/5]
[0 1/5 | 3/5]
3. Perform the required row operations to get a leading coefficient of 1 in the second row.
- Divide the second row by 1/5: (1/5)*R2 -> R2
- The augmented matrix becomes:
[1 -4/5 | 12/5]
[0 1 | 3/5]
4. Perform the required row operations to eliminate the coefficient of y in the first row.
- Multiply the second row by 4/5 and add it to the first row: (4/5)*R2 + R1 -> R1
- The augmented matrix becomes:
[1 0 | 12/5 - (4/5)*(3/5)]
[0 1 | 3/5]
5. Simplify the augmented matrix.
- Perform the necessary calculations to simplify the entries:
[1 0 | 12/5 - 12/25]
[0 1 | 3/5]
6. Final step, interpret the augmented matrix.
- The simplified augmented matrix represents the solution to the system of equations. The first row tells us that x = 12/5 - 12/25 = 48/25 - 12/25 = 36/25. The second row tells us that y = 3/5.
The solution to the system of equations is x = 36/25 and y = 3/5.