Solve using augmented matrices.
x + y = 1
3x – y = 7
Please help, I need a step-by-step explination. I dont understand it at all...
Please help, I really need help to even understand the subject. I am really lost...
is that solving using augmented matrices?
no, that is substitution.
Looks like you need to review elimination and augmented matrices.
I tried but I didn't understand it. I was saying that
Your answer didn't help at all. I already know how to solve using elimination and substitution. But thanks anyway...
To solve the system of equations using augmented matrices, you need to perform the following steps:
Step 1: Write the augmented matrix:
The augmented matrix is formed by writing the coefficients of the variables along with the constants in a matrix form. For the given system of equations:
x + y = 1 is equivalent to 1x + 1y = 1
3x – y = 7 is equivalent to 3x - 1y = 7
The augmented matrix is:
[ 1 1 | 1 ]
[ 3 -1 | 7 ]
Step 2: Perform row operations to reduce the augmented matrix to row-echelon form:
The goal of row operations is to transform the augmented matrix into row-echelon form, where the leading entry (the leftmost nonzero entry) in each row is 1, and all the entries below it are zero.
To achieve this, we can use Gaussian elimination. The row operations allowed are:
- Interchanging two rows
- Multiplying a row by a nonzero constant
- Adding a multiple of one row to another row
Here's how we can perform the row operations:
a) Multiply row 1 by -3 and add it to row 2:
[ 1 1 | 1 ]
+ [-3 -3 | -3 ]
____________________
[-2 -2 | -2 ]
The resulting matrix is:
[ 1 1 | 1 ]
[ 0 -4 | -2 ]
b) Divide row 2 by -4:
[ 1 1 | 1 ]
[ 0 -4 | -2 ]
____________________
[ 1 -1 | 1/2 ]
The resulting matrix is:
[ 1 1 | 1 ]
[ 0 -1 | -1/2 ]
Step 3: Continue performing row operations to obtain the reduced row-echelon form:
The remaining row operation is to multiply row 2 by -1:
[ 1 1 | 1 ]
[ 0 -1 | -1/2 ]
____________________
[ 1 -1 | -1/2 ]
The resulting matrix is:
[ 1 1 | 1 ]
[ 0 1 | 1/2 ]
This matrix is now in reduced row-echelon form.
Step 4: Interpret the augmented matrix back into equations:
From the reduced row-echelon form, we can write the following equations:
x + y/2 = 1
y = 1/2
Step 5: Solve for the variables:
Substituting the value of y = 1/2 into the first equation, we get:
x + 1/2 = 1
x = 1 - 1/2
x = 1/2
Therefore, the solution to the system of equations is x = 1/2 and y = 1/2.
top: y = 1-x
so, use that in the bottom:
3x - (1-x) = 7
4x-1 = 7
x = 2
y = 1-x = -1