Math / Algebra

QuestionQuiz 9: 6.1, 6.3 and 6.5 Question 8 of 8 (1 point) | Question Attempt: 1 of 1 Time Remaining: $38: 27$ Antonina
Solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. Write all numbers as integers or simplified fractions. $\begin{array}{l} 9 x-4 y=-16 \\ 8 x-3 y=-19 \end{array}$
Part: $0 / 2$
Part 1 of 2
Evaluate the determinants $D, D_{v}$ , and $D_{v}$ .

Studdy Solution

STEP 1

What is this asking? We need to solve a system of two equations with two variables ( $x$ and $y$ ) using Cramer's rule, which involves calculating determinants, or if Cramer's rule doesn't work, use another method. Watch out! Cramer's rule doesn't work if the determinant of the coefficient matrix is zero, so we need to check for that!

STEP 2

1. Calculate D
2. Calculate Dx
3. Calculate Dy
4. Solve for x and y

STEP 3

Let's build our coefficient matrix from the coefficients of $x$ and $y$ in our equations.
This will be a $2 \times 2$ matrix: $\begin{bmatrix} 9 & -4 \\ 8 & -3 \end{bmatrix}$

STEP 4

The determinant D is calculated as: $D = (9 \cdot -3) - (-4 \cdot 8)$ $D = -27 - (-32)$ $D = -27 + 32$ $D = \textbf{5}$ Since $D$ is not zero, we can use Cramer's rule!
Woohoo!

STEP 5

To find Dx, we replace the x coefficients in the coefficient matrix with the constants from the right-hand side of the equations: $\begin{bmatrix} -16 & -4 \\ -19 & -3 \end{bmatrix}$

STEP 6

Now, we calculate the determinant Dx: $D_x = (-16 \cdot -3) - (-4 \cdot -19)$ $D_x = 48 - 76$ $D_x = \textbf{-28}$

STEP 7

To find Dy, we replace the y coefficients in the coefficient matrix with the constants from the right-hand side of the equations: $\begin{pmatrix} 9 & -16 \\ 8 & -19 \end{pmatrix}$

STEP 8

Now, we calculate the determinant Dy: $D_y = (9 \cdot -19) - (-16 \cdot 8)$ $D_y = -171 - (-128)$ $D_y = -171 + 128$ $D_y = \textbf{-43}$

STEP 9

Now, we can find $x$ by dividing Dx by D: $x = \frac{D_x}{D} = \frac{-28}{5}$

STEP 10

And we can find $y$ by dividing Dy by D: $y = \frac{D_y}{D} = \frac{-43}{5}$

STEP 11

$x = -\frac{28}{5}$ and $y = -\frac{43}{5}$

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Studdy Solution

STEP 1

STEP 2

1. Calculate D
2. Calculate Dx
3. Calculate Dy
4. Solve for x and y

STEP 3

Let's build our coefficient matrix from the coefficients of $x$ and $y$ in our equations.
This will be a $2 \times 2$ matrix: $\begin{bmatrix} 9 & -4 \\ 8 & -3 \end{bmatrix}$

STEP 4

The determinant D is calculated as: $D = (9 \cdot -3) - (-4 \cdot 8)$ $D = -27 - (-32)$ $D = -27 + 32$ $D = \textbf{5}$ Since $D$ is not zero, we can use Cramer's rule!
Woohoo!

STEP 5

To find Dx, we replace the x coefficients in the coefficient matrix with the constants from the right-hand side of the equations: $\begin{bmatrix} -16 & -4 \\ -19 & -3 \end{bmatrix}$

STEP 6

Now, we calculate the determinant Dx: $D_x = (-16 \cdot -3) - (-4 \cdot -19)$ $D_x = 48 - 76$ $D_x = \textbf{-28}$

STEP 7

To find Dy, we replace the y coefficients in the coefficient matrix with the constants from the right-hand side of the equations: $\begin{pmatrix} 9 & -16 \\ 8 & -19 \end{pmatrix}$

STEP 8

Now, we calculate the determinant Dy: $D_y = (9 \cdot -19) - (-16 \cdot 8)$ $D_y = -171 - (-128)$ $D_y = -171 + 128$ $D_y = \textbf{-43}$

STEP 9

Now, we can find $x$ by dividing Dx by D: $x = \frac{D_x}{D} = \frac{-28}{5}$

STEP 10

And we can find $y$ by dividing Dy by D: $y = \frac{D_y}{D} = \frac{-43}{5}$

STEP 11

$x = -\frac{28}{5}$ and $y = -\frac{43}{5}$

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Studdy Solution

STEP 1

STEP 2

1. Calculate D2. Calculate Dx3. Calculate Dy4. Solve for x and y

STEP 3

Let's **build our coefficient matrix** from the coefficients of xxx and yyy in our equations.This will be a 2×22 \times 22×2 matrix: [9−48−3]\begin{bmatrix} 9 & -4 \\ 8 & -3 \end{bmatrix}[98​−4−3​]

STEP 4

The **determinant D** is calculated as: D=(9⋅−3)−(−4⋅8)D = (9 \cdot -3) - (-4 \cdot 8)D=(9⋅−3)−(−4⋅8) D=−27−(−32)D = -27 - (-32)D=−27−(−32)D=−27+32D = -27 + 32D=−27+32D=5D = \textbf{5}D=5Since DDD is not zero, we can use Cramer's rule!Woohoo!

STEP 5

To find **Dx**, we replace the *x* coefficients in the coefficient matrix with the constants from the right-hand side of the equations: [−16−4−19−3]\begin{bmatrix} -16 & -4 \\ -19 & -3 \end{bmatrix}[−16−19​−4−3​]

STEP 6

Now, we calculate the **determinant Dx**: Dx=(−16⋅−3)−(−4⋅−19)D_x = (-16 \cdot -3) - (-4 \cdot -19)Dx​=(−16⋅−3)−(−4⋅−19) Dx=48−76D_x = 48 - 76Dx​=48−76Dx=-28D_x = \textbf{-28}Dx​=-28

STEP 7

To find **Dy**, we replace the *y* coefficients in the coefficient matrix with the constants from the right-hand side of the equations: (9−168−19)\begin{pmatrix} 9 & -16 \\ 8 & -19 \end{pmatrix}(98​−16−19​)

STEP 8

Now, we calculate the **determinant Dy**: Dy=(9⋅−19)−(−16⋅8)D_y = (9 \cdot -19) - (-16 \cdot 8)Dy​=(9⋅−19)−(−16⋅8) Dy=−171−(−128)D_y = -171 - (-128)Dy​=−171−(−128)Dy=−171+128D_y = -171 + 128Dy​=−171+128Dy=-43D_y = \textbf{-43}Dy​=-43

STEP 9

Now, we can find xxx by dividing **Dx** by **D**: x=DxD=−285x = \frac{D_x}{D} = \frac{-28}{5}x=DDx​​=5−28​

STEP 10

And we can find yyy by dividing **Dy** by **D**: y=DyD=−435y = \frac{D_y}{D} = \frac{-43}{5}y=DDy​​=5−43​

STEP 11

x=−285x = -\frac{28}{5}x=−528​ and y=−435y = -\frac{43}{5}y=−543​

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1. Calculate D
2. Calculate Dx
3. Calculate Dy
4. Solve for x and y

Let's build our coefficient matrix from the coefficients of $x$ and $y$ in our equations.
This will be a $2 \times 2$ matrix: $\begin{bmatrix} 9 & -4 \\ 8 & -3 \end{bmatrix}$

The determinant D is calculated as: $D = (9 \cdot -3) - (-4 \cdot 8)$ $D = -27 - (-32)$ $D = -27 + 32$ $D = \textbf{5}$ Since $D$ is not zero, we can use Cramer's rule!
Woohoo!

To find Dx, we replace the x coefficients in the coefficient matrix with the constants from the right-hand side of the equations: $\begin{bmatrix} -16 & -4 \\ -19 & -3 \end{bmatrix}$

Now, we calculate the determinant Dx: $D_x = (-16 \cdot -3) - (-4 \cdot -19)$ $D_x = 48 - 76$ $D_x = \textbf{-28}$

To find Dy, we replace the y coefficients in the coefficient matrix with the constants from the right-hand side of the equations: $\begin{pmatrix} 9 & -16 \\ 8 & -19 \end{pmatrix}$

Now, we calculate the determinant Dy: $D_y = (9 \cdot -19) - (-16 \cdot 8)$ $D_y = -171 - (-128)$ $D_y = -171 + 128$ $D_y = \textbf{-43}$

Now, we can find $x$ by dividing Dx by D: $x = \frac{D_x}{D} = \frac{-28}{5}$

And we can find $y$ by dividing Dy by D: $y = \frac{D_y}{D} = \frac{-43}{5}$

$x = -\frac{28}{5}$ and $y = -\frac{43}{5}$