Solve the system using an augmented matrix. Show all work.
-x - 3y = -17
2x - 6y = -26
Please help me! I don’t know how to do this. Can you show how to get it and the solution?
Of course! I'll walk you through solving this system of equations using an augmented matrix.
Step 1: Write down the system of equations:
-x - 3y = -17 ...(Equation 1)
2x - 6y = -26 ...(Equation 2)
Step 2: Convert the system into an augmented matrix. The coefficients of the variables x and y, along with the constants on the right side of the equations, will form the augmented matrix.
Start with the coefficients and constants from Equation 1 in the first row, and Equation 2 in the second row. The matrix will look like this:
|-1 -3 -17|
| 2 -6 -26|
Step 3: Perform row operations to put the augmented matrix into row-echelon form or reduced row-echelon form. The goal is to have zeros below the main diagonal.
We'll use Gaussian elimination to achieve this. Let's start by getting a 2 in the top left corner as the leading coefficient:
R1 + 2R2 → R1:
|-1 -3 -17|
| 0 0 0|
We now have a row of zeros in the second row. This indicates that one of the equations is dependent on the other. We only have one equation left. Let's solve it:
-x - 3y = -17
Rearrange equation 1:
-x = 3y - 17
x = -3y + 17
Now, we can freely choose any value for y, and then find the corresponding value for x:
Let's set y = t (a parameter).
Substitute y = t into x = -3y + 17:
x = -3(t) + 17
x = -3t + 17
Therefore, the general solution to the system of equations is:
x = -3t + 17
y = t
The system has infinitely many solutions, represented by the parameter t.