Question

Apply Crammer's Rule to find the solution to the following quations .2x + 3y = 1, 3x + y = 5​

Answer 1

The solution to the equation system given is:

• x = 2
• y = -1

Explanation:

First, we must know the equations given:

1. 2x + 3y = 1
2. 3x + y = 5​

Following Crammer's Rule, we have the matrix form:

`\left[\begin{array}{ccc}2&3\\3&1\end{array}\right] =\left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}1\\5\end{array}\right]`

Now we solve using the determinants:

`x=\frac{\left[\begin{array}{ccc}1&3\\5&1\end{array}\right]}{\left[\begin{array}{ccc}2&3\\3&1\end{array}\right] } =((1*1)-(5*3))/((2*1)-(3*3)) = (1-15)/(2-9) =(-14)/(-7) = 2`

`y=\frac{\left[\begin{array}{ccc}2&1\\3&5\end{array}\right]}{\left[\begin{array}{ccc}2&3\\3&1\end{array}\right] } =((2*5)-(3*1))/((2*1)-(3*3))=(10-3)/(2-9) =(7)/(-7)=-1`

Now, we can find the answer which is x= 2 and y= -1, we can replace these values in the equation to confirm the results are right, with the first equation:

• 2x + 3y = 1
• 2(2) + 3(-1)= 1
• 4 - 3 = 1
• 1 = 1

And, with the second equation:

• 3x + y = 5​
• 3(2) + (-1) = 5
• 6 - 1 = 5
• 5 = 5