how do u solve for x and y of
x+2y=7
2x+y=8
using matrix without using a calculator
To solve the system of equations using matrices without a calculator, you can represent the system in matrix form and then apply matrix operations to find the values of x and y.
Step 1: Write the system of equations in matrix form:
| 1 2 | | x | | 7 |
| | | | = | |
| 2 1 | | y | | 8 |
Step 2: Create the augmented matrix by combining the coefficient matrices on the left-hand side and the constants on the right-hand side:
| 1 2 | 7 |
| | |
| 2 1 | 8 |
Step 3: Apply elementary row operations to the augmented matrix to convert it to reduced row-echelon form (RREF). The goal is to have a matrix where the coefficients are all zeros except for the leading coefficient in each row.
First, let's make the leading coefficient in the first row equal to 1 by dividing the entire row by 1:
| 1 2 | 7 |
| | |
| 2 1 | 8 |
Next, let's eliminate the coefficient of 2 in the second row by subtracting 2 times the first row from the second row:
| 1 2 | 7 |
| -2 -3 | -6 |
| | |
| 2 1 | 8 |
Finally, let's eliminate the coefficient of 2 in the third row by subtracting 2 times the first row from the third row:
| 1 2 | 7 |
| -2 -3 | -6 |
| -2 -3 | -6 |
Now our matrix is in RREF form.
Step 4: Read off the solutions from the augmented matrix:
x = 7
-2x - 3y = -6
y = (6 - (-2*7))/(-3)
= (6 + 14)/(-3)
= 20/(-3)
= -20/3
So, the solution to the system of equations is x = 7 and y = -20/3.