Math Snap

STEP 1

What is this asking?
We need to find the numerator of the fraction when solving for $x$ in a system of two equations using Cramer's Rule.
Watch out!
Don't forget to use the correct coefficients when setting up the determinant for the numerator!

STEP 2

1. Set up the system of equations.
2. Calculate the determinant for the numerator.

STEP 3

Alright, let's begin by clearly writing out our system of equations:
$$\begin{array}{l} 2x + 3y = 5 \\ x - 4y = -3 \end{array}$$

STEP 4

Remember, Cramer's rule helps us solve for $x$ and $y$ using determinants.
It's like a shortcut to find the solutions!

STEP 5

When using Cramer's Rule to solve for $x$, the numerator of the $x$ solution is a special determinant.
We replace the coefficients of $x$ in the system of equations with the constant terms.

STEP 6

The coefficients of $x$ are $\mathbf{2}$ and $\mathbf{1}$.
The constant terms are $\mathbf{5}$ and $\mathbf{-3}$.
So, we replace the $x$ coefficients with the constant terms.
The coefficients of $y$ remain unchanged.

STEP 7

This gives us the determinant:
$$\begin{vmatrix} 5 & 3 \\ -3 & -4 \end{vmatrix}$$

STEP 8

Now, let's calculate this determinant!
Remember, it's the product of the main diagonal minus the product of the off-diagonal.
$$(\mathbf{5} \cdot (\mathbf{-4})) - (3 \cdot (\mathbf{-3}))$$ $$\mathbf{-20} - (\mathbf{-9})$$

STEP 9

Remember, subtracting a negative number is the same as adding its positive counterpart.
So, we have:
$$\mathbf{-20} + \mathbf{9} = \mathbf{-11}$$ So, the determinant of the numerator is $\mathbf{-11}$!

The value of $x$ that satisfies the system of equations is $x = \frac{\mathbf{-11}}{\begin{vmatrix} 2 & 3 \\ 1 & -4 \end{vmatrix}}$.
Therefore, the missing value is $\mathbf{-11}$.