Solve the system using an augmented matrix. Show all work.
{−x−3y=−17
2x−6y=−26
The system of equations can be represented as an augmented matrix:
\[
\begin{array}{ccc|c}
-1 & -3 & -17 \\
2 & -6 & -26 \\
\end{array}
\]
To solve the system using row operations, we will first perform row operations to get a leading 1 in the first column and zeros below it.
\[
\begin{array}{ccc|c}
-1 & -3 & -17 \\
2 & -6 & -26 \\
\end{array}
\]
Multiply the first row by -2 and add it to the second row:
\[
\begin{array}{ccc|c}
-1 & -3 & -17 \\
0 & 0 & 8 \\
\end{array}
\]
This means the second row represents the equation 0x + 0y = 8, which simplifies to 0 = 8. Since this equation is not true, the system has no solution.
Therefore, the system is inconsistent and there is no solution.
To solve the given system of equations using an augmented matrix, we will first write the equations in the form of ax + by = c.
The given system of equations is:
1) -x - 3y = -17
2) 2x - 6y = -26
To write these equations in the form ax + by = c, we multiply equation 1 by -2:
-2(-x - 3y) = -2(-17)
2x + 6y = 34
So the system becomes:
1) 2x + 6y = 34
2) 2x - 6y = -26
Now, we can create the augmented matrix for the system by listing the coefficients of x, y, and the constant terms:
[ 2 6 | 34 ]
[ 2 -6 | -26 ]
To solve the system using row operations, we'll start by multiplying Row 2 by -1/2:
[ 2 6 | 34 ]
[ -1 3 | 13 ]
Next, we'll replace Row 2 with the sum of Row 2 and Row 1:
[ 2 6 | 34 ]
[ 1 9 | 47 ]
Now, we'll replace Row 1 with Row 1 - 2 * Row 2:
[ 0 -12 | -60 ]
[ 1 9 | 47 ]
Next, we'll divide Row 2 by 3:
[ 0 -12 | -60 ]
[ 1 3 | 15 ]
Finally, we'll replace Row 1 with Row 1 + 4 * Row 2:
[ 4 0 | 0 ]
[ 1 3 | 15 ]
Now, we can see that the augmented matrix is in the form:
[ x1 | 0 ]
[ x2 | c ]
From the augmented matrix, we can read off the solution to the system:
x1 = 0
x2 = 15
Therefore, the solution to the given system of equations is x = 0 and y = 15.
To solve the system using an augmented matrix, we'll need to first write the system of equations in matrix form.
The given system of equations is:
- x - 3y = -17 --- (1)
2x - 6y = -26 --- (2)
To write it in matrix form, we'll set up the augmented matrix [A | B] where A is the coefficient matrix and B is the matrix containing the constant terms.
The coefficient matrix (A) is obtained by taking the coefficients of x and y from each equation:
A = | -1 -3 |
| 2 -6 |
The constant matrix (B) is formed by taking the constant terms from each equation:
B = | -17 |
| -26 |
Now, let's set up the augmented matrix [A | B] using A and B:
[A | B] = | -1 -3 | -17 |
| 2 -6 | -26 |
To solve the system using the augmented matrix, we can perform row operations to transform the augmented matrix into row-echelon form.
Row operations:
1. Multiply a row by a non-zero scalar.
2. Add or subtract rows.
3. Switch rows.
We want to transform the augmented matrix into the following form:
[R1 | c1]
[0 R2 | c2]
where R1 and R2 are rows and c1, c2 are constants.
Now, let's perform the row operations to simplify the augmented matrix:
First, we'll multiply R1 by -2 so that we can eliminate x in R2:
[R1 | -2B] = | 2 6 | 34 | => R1 * -2 + R2 -> R2
[0 R2 | c2] = | 2 -6 | -26 |
[R1 | -2B] = | 2 6 | 34 |
[0 R2 | c2] = | 0 0 | -18 |
Next, let's simplify R2 further by multiplying it by -1/18:
[R1 | -2B] = | 2 6 | 34 |
[0 R2 | c2] = | 0 0 | 1 |
[R1 | -2B] = | 2 6 | 34 |
[0 R2 | c2] = | 0 0 | 1 |
Since R2 is in the form [0 0 | 1], this means that 0x + 0y = 1, which is a contradiction. Therefore, the system of equations is inconsistent, meaning there is no solution that satisfies both equations.
In conclusion, the given system of equations is inconsistent and has no solution.