11.8 Solving Systems with Cramer's Rule

Use Cramer’s rule to solve the system of equations.

$\left\{\begin{matrix} - 2 x + 3 y = 3 \\ x + 3 y = 12 \end{matrix}\right$

Find the determinant of the given matrix.

Solve the following $2 \times 2$ system using Cramer’s Rule.

Use Cramer’s Rule to solve the 2 × 2 system of equations.

Find the determinant of the 3 × 3 matrix given

Find the determinant of the 3 × 3 matrix.

Find the solution to the given 3 × 3 system using Cramer’s Rule.

Use Cramer’s Rule to solve the 3 × 3 matrix.

Illustrate each of the properties of determinants.

Find the solution to the given 3 × 3 system.

Explain why we can always evaluate the determinant of a square matrix.

Explain what it means in terms of an inverse for a matrix to have a 0 determinant.

Find the determinant.

$\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} 2 & - 5 \\ - 1 & 6 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} 1 & 0 \\ 3 & - 4 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} 10 & 0.2 \\ 5 & 0.1 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} - 2 & - 3 \\ 3.1 & 4 , 000 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 3 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} - 2 & 1 & 4 \\ - 4 & 2 & - 8 \\ 2 & - 8 & - 3 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} 5 & 1 & - 1 \\ 2 & 3 & 1 \\ 3 & - 6 & - 3 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} 2 & - 1.6 & 3.1 \\ 1.1 & 3 & - 8 \\ - 9.3 & 0 & 2 \end{vmatrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}2 x - 3 y = −1 \\ 4 x + 5 y = 9\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}6 x - 3 y = 2 \\ - 8 x + 9 y = −1\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}4 x + 3 y = 23 \\ 2 x - y = −1\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}4 x - 3 y = −3 \\ 2 x + 6 y = −4\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}4 x + 10 y = 180 \\ - 3 x - 5 y = −105\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}x + 2 y - 4 z = - 1 \\ 7 x + 3 y + 5 z = 26 \\ - 2 x - 6 y + 7 z = - 6\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}4 x + 5 y - z = −7 \\ −2 x - 9 y + 2 z = 8 \\ 5 y + 7 z = 21\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}4 x - 2 y + 3 z = 6 \\ - 6 x + y = - 2 \\ 2 x + 7 y + 8 z = 24\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}13 x - 17 y + 16 z = 73 \\ - 11 x + 15 y + 17 z = 61 \\ 46 x + 10 y - 30 z = - 18\end{matrix}$

Solve the system of linear equations using Cramer’s Rule.

$\begin{matrix}4 x - 6 y + 8 z = 10 \\ - 2 x + 3 y - 4 z = - 5 \\ x + y + z = 1\end{matrix}$

Use the determinant function on a graphing utility.

$\begin{vmatrix} 1 & 0 & 8 & 9 \\ 0 & 2 & 1 & 0 \\ 1 & 0 & 3 & 0 \\ 0 & 2 & 4 & 3 \end{vmatrix}$

Use the determinant function on a graphing utility.

$\begin{vmatrix} \frac{1}{2} & 1 & 7 & 4 \\ 0 & \frac{1}{2} & 100 & 5 \\ 0 & 0 & 2 & 2,000 \\ 0 & 0 & 0 & 2 \end{vmatrix}$

Create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.

Two numbers add up to 56. One number is 20 less than the other.

Create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.

Three numbers add up to 106. The first number is 3 less than the second number. The third number is 4 more than the first number.

Create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

You invest $10,000 into two accounts, which receive 8% interest and 5% interest. At the end of a year, you had $10,710 in your combined accounts. How much was invested in each account?

Create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

A theater needs to know how many adult tickets and children tickets were sold out of the 1,200 total tickets. If children’s tickets are $5.95, adult tickets are $11.15, and the total amount of revenue was $12,756, how many children’s tickets and adult tickets were sold?

Create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were 10 gal total. Additionally, you kept your receipt, and know the total amount s…

Create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

Your garden produced two types of tomatoes, one green and one red. The red weigh 10 oz, and the green weigh 4 oz. You have 30 tomatoes, and a total weight of 13 lb, 14 oz. How many of each type of tomato do you have?

Create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

At the same market, the three most popular fruits make up 37% of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.

Create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

A movie theatre sold tickets to three movies. The tickets to the first movie were $5, the tickets to the second movie were $11, and the third movie was $12. 100 tickets were sold to the first movie. The total number of tickets sold was 642, for a total revenue of $6,774. How m…

For the “hiking” mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. If there is the same amount of almonds as cashews, how many of each item is in the trail mix?

Determine whether the ordered pair is a solution to the system of equations.

$\begin{matrix}3 x - y = 4 \\ x + 4 y = - 3\end{matrix}$ and $\left(- 1 , 1\right)$

Use substitution to solve the system of equations.

$\begin{matrix}10 x + 5 y = −5 \\ 3 x - 2 y = −12\end{matrix}$

Use substitution to solve the system of equations.

$\begin{matrix}5 x + 6 y = 14 \\ 4 x + 8 y = 8\end{matrix}$

Use addition to solve the system of equations.

$\begin{matrix}3 x + 4 y = 2 \\ 9 x + 12 y = 3\end{matrix}$

Write a system of equations to solve each problem. Solve the system of equations.

A factory has a cost of production $C \left(x\right) = 150 x + 15 , 000$ and a revenue function $R \left(x\right) = 200 x .$ What is the break-even point?

Solve the system of three equations using substitution or addition.

$\begin{matrix}0.5 x - 0.5 y = 10 \\ - 0.2 y + 0.2 x = 4 \\ 0.1 x + 0.1 z = 2\end{matrix}$

Solve the system of three equations using substitution or addition.

$\begin{matrix}x + y + z = 1 \\ 2 x + 2 y + 2 z = 1 \\ 3 x + 3 y = 2\end{matrix}$

Solve the system of three equations using substitution or addition.

$\begin{matrix}3 x + 2 y - z = −10 \\ x - y + 2 z = 7 \\ - x + 3 y + z = −2\end{matrix}$

Solve the system of three equations using substitution or addition.

$\begin{matrix}2 x - 3 y + z = 0 \\ 2 x + 4 y - 3 z = 0 \\ 6 x - 2 y - z = 0\end{matrix}$

Write a system of equations to solve each problem. Solve the system of equations.

Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?

Solve the system of nonlinear equations.

$\begin{matrix}y = x^{2} - 7 \\ y = 5 x - 13\end{matrix}$

Solve the system of nonlinear equations.

$\begin{matrix}x^{2} + y^{2} = 16 \\ y = x - 8\end{matrix}$

Solve the system of nonlinear equations.

$\begin{matrix}x^{2} + y^{2} = 4 \\ y - x^{2} = 3\end{matrix}$

Decompose into partial fractions.

$\frac{- 2 x + 6}{x^{2} + 3 x + 2}$

Decompose into partial fractions.

$\frac{7 x + 20}{x^{2} + 10 x + 25}$

Decompose into partial fractions.

$\frac{- x^{2} + 36 x + 70}{x^{3} - 125}$

Decompose into partial fractions.

$\frac{x^{3} - 4 x^{2} + 3 x + 11}{\left(x^{2} - 2\right)^{2}}$

$- 4 A$

$B + C$

$B A$

$C B$

$E D$

$C E$

Write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

$\begin{bmatrix} 1 & 0 & −3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \left|\right. 7 \\ −5 \\ 0 \end{bmatrix}$

Write the augmented matrix from the system of linear equations.

$\begin{matrix}- 2 x + 2 y + z = 7 \\ 2 x - 8 y + 5 z = 0 \\ 19 x - 10 y + 22 z = 3\end{matrix}$

Write the augmented matrix from the system of linear equations.

$\begin{matrix}x + 3 z = 12 \\ - x + 4 y = 0 \\ y + 2 z = - 7\end{matrix}$

Solve the system of linear equations using Gaussian elimination.

$\begin{matrix}3 x - 4 y = 1 \\ - 6 x + 8 y = 6\end{matrix}$

Solve the system of linear equations using Gaussian elimination.

$\begin{matrix}2 x + 3 y + 2 z = 1 \\ - 4 x - 6 y - 4 z = - 2 \\ 10 x + 15 y + 10 z = 0\end{matrix}$

Find the inverse of the matrix.

$\begin{bmatrix} - 0.2 & 1.4 \\ 1.2 & - 0.4 \end{bmatrix}$

Find the inverse of the matrix.

$\begin{bmatrix} 12 & 9 & - 6 \\ - 1 & 3 & 2 \\ - 4 & - 3 & 2 \end{bmatrix}$

Find the solutions by computing the inverse of the matrix.

$\begin{matrix}0.3 x - 0.1 y = - 10 \\ - 0.1 x + 0.3 y = 14\end{matrix}$

Find the solutions by computing the inverse of the matrix.

$\begin{matrix}4 x + 3 y - 3 z = - 4.3 \\ 5 x - 4 y - z = - 6.1 \\ x + z = - 0.7\end{matrix}$

Write a system of equations to solve each problem. Solve the system of equations.

Students were asked to bring their favorite fruit to class. 90% of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were 5% more popular than bananas, what are the percentages of each individual fruit?

Find the determinant.

$\begin{vmatrix} 100 & 0 \\ 0 & 0 \end{vmatrix}$

Find the determinant.

$\begin{vmatrix} - 1 & 4 & 3 \\ 0 & 2 & 3 \\ 0 & 0 & - 3 \end{vmatrix}$

Use Cramer’s Rule to solve the linear systems of equations.

$\begin{matrix}4 x - 2 y = 23 \\ - 5 x - 10 y = - 35\end{matrix}$

Use Cramer’s Rule to solve the linear systems of equations.

$\begin{matrix}- 0.5 x + 0.1 y = 0.3 \\ - 0.25 x + 0.05 y = 0.15\end{matrix}$

Use Cramer’s Rule to solve the linear systems of equations.

$\begin{matrix}4 x - 3 y + 5 z = - \frac{5}{2} \\ 7 x - 9 y - 3 z = \frac{3}{2} \\ x - 5 y - 5 z = \frac{5}{2}\end{matrix}$

Is the following ordered pair a solution to the system of equations?

$\begin{matrix} \\ \begin{matrix}- 5 x - y = 12 \\ x + 4 y = 9\end{matrix}\end{matrix}$ with $\left(- 3 , 3\right)$

Solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{aligned} \\ - \frac{1}{2} x - 4 y = 4 \\ 2 x + 16 y = 2\end{aligned}$

Solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{matrix}4 x - 6 y - 2 z = \frac{1}{10} \\ x - 7 y + 5 z = - \frac{1}{4} \\ 3 x + 6 y - 9 z = \frac{6}{5}\end{matrix}$

Solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{matrix}5 x - 4 y - 3 z = 0 \\ 2 x + y + 2 z = 0 \\ x - 6 y - 7 z = 0\end{matrix}$

Solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

$\begin{matrix}y^{2} + x^{2} = 25 \\ y^{2} - 2 x^{2} = 1\end{matrix}$

Write the partial fraction decomposition.

$\frac{13 x + 2}{\left(3 x + 1\right)^{2}}$

Perform the given matrix operations.

$5 \begin{bmatrix} 4 & 9 \\ - 2 & 3 \end{bmatrix} + \frac{1}{2} \begin{bmatrix} - 6 & 12 \\ 4 & - 8 \end{bmatrix}$

Perform the given matrix operations.

$\begin{bmatrix} \frac{1}{2} & \frac{1}{3} \\ \frac{1}{4} & \frac{1}{5} \end{bmatrix}^{- 1}$

Perform the given matrix operations.

$\det \begin{vmatrix} \frac{1}{2} & - \frac{1}{2} & 0 \\ - \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & 0 \end{vmatrix}$

Perform the given matrix operations.

Rewrite the system of linear equations as an augmented matrix.

Use Gaussian elimination to solve the systems of equations.

$\begin{matrix}x - 6 y = 4 \\ 2 x - 12 y = 0\end{matrix}$

Use the inverse of a matrix to solve the systems of equations.

$\begin{matrix}4 x - 5 y = - 50 \\ - x + 2 y = 80\end{matrix}$

Use Cramer’s Rule to solve the systems of equations.

$\begin{matrix}200 x - 300 y = 2 \\ 400 x + 715 y = 4\end{matrix}$

Solve using a system of linear equations.

A factory producing cell phones has the following cost and revenue functions: $C \left(x\right) = x^{2} + 75 x + 2 , 688$ and $R \left(x\right) = x^{2} + 160 x .$ What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.