Math / Algebra

QuestionComplete the sentence below by replacing ? with an appropriate value. Using Cramer's Rule, the value of $x$ that satisfies the system of equations $\left\{\begin{array}{l}2 x+3 y=5 \\ x-4 y=-3\end{array}\right.$ is $x=\frac{?}{\left|\begin{array}{rr}2 & 3 \\ 1 & -4\end{array}\right|}$

Studdy Solution

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 $\mathbf{-11}$ !

STEP 10

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 $\mathbf{-11}$ .

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QuestionComplete the sentence below by replacing ? with an appropriate value. Using Cramer's Rule, the value of $x$ that satisfies the system of equations $\left\{\begin{array}{l}2 x+3 y=5 \\ x-4 y=-3\end{array}\right.$ is $x=\frac{?}{\left|\begin{array}{rr}2 & 3 \\ 1 & -4\end{array}\right|}$

Studdy Solution

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 $\mathbf{-11}$ !

STEP 10

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 $\mathbf{-11}$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the numerator of the fraction when solving for xxx 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: 2x+3y=5x−4y=−3\begin{array}{l} 2x + 3y = 5 \\ x - 4y = -3 \end{array}2x+3y=5x−4y=−3​

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

This gives us the determinant: ∣53−3−4∣\begin{vmatrix} 5 & 3 \\ -3 & -4 \end{vmatrix}∣∣​5−3​3−4​∣∣​

STEP 8

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

STEP 9

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

STEP 10

The value of xxx that satisfies the system of equations is x=−11∣231−4∣ x = \frac{\mathbf{-11}}{\begin{vmatrix} 2 & 3 \\ 1 & -4 \end{vmatrix}} x=∣∣​21​3−4​∣∣​−11​.Therefore, the missing value is −11\mathbf{-11}−11.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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!

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

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

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

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.

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.

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

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

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 $\mathbf{-11}$ .