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Not enough information. Not enough pictures. Any additional feedback? Submit feedback. 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.
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? What percentage does each vegetable have in the market share? 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. Three bands performed at a concert venue.
If the first band had 40 more audience members than the second band, how many tickets were sold for each band? A movie theatre sold tickets to three movies. How many tickets for each movie were sold? This year, the same age groups made up Determine the prison population percentage for each age group last year. There are now 6, prisoners.
Originally, there were more in the 30—39 age group than the 20—29 age group. Determine the prison population for each age group last year. For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews.
The nutritional information for these items is shown in Figure. The total number of carbohydrates is g, and the total amount of fat is If there are more pieces of cashews than cranberries, how many of each item is in the trail mix?
If there is the same amount of almonds as cashews, how many of each item is in the trail mix? If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix? For the following exercises, determine whether the ordered pair is a solution to the system of equations. For the following exercises, use substitution to solve the system of equations.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations. A factory has a cost of production and a revenue function What is the break-even point? A performer charges where is the total number of attendees at a show. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?
For the following exercises, solve the system of three equations using substitution or addition. Three odd numbers sum up to The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers? A local theatre sells out for their show. For the following exercises, perform the requested operations on the given matrices. For the following exercises, write the system of linear equations from the augmented matrix.
Indicate whether there will be a unique solution. For the following exercises, write the augmented matrix from the system of linear equations. For the following exercises, solve the system of linear equations using Gaussian elimination. For the following exercises, find the solutions by computing the inverse of the matrix. Students were asked to bring their favorite fruit to class. A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies.
How many brownies and how many cookies were sold? For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists. If what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse? For the following exercises, use Gaussian elimination to solve the systems of equations. For the following exercises, use the inverse of a matrix to solve the systems of equations.
A factory producing cell phones has the following cost and revenue functions: and What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit. In one day, three times as many children as adults attended. How many of each type of ticket was sold? Skip to content Systems of Equations and Inequalities. Know the properties of determinants. Consider a system of two linear equations in two variables. Figure 1. Understanding Properties of Determinants There are many properties of determinants.
Properties of Determinants If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal. When two rows are interchanged, the determinant changes sign. If either two rows or two columns are identical, the determinant equals zero.
If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero. The determinant of an inverse matrix is the reciprocal of the determinant of the matrix If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.
Illustrating Properties of Determinants Illustrate each of the properties of determinants. Key Concepts The determinant for is See Figure. Solutions are See Figure. Add the three diagonal entries upper left to lower right and subtract the three diagonal entries lower left to upper right. See Figure and Figure. Certain properties of determinants are useful for solving problems.
For example: If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal. Section Exercises Verbal Explain why we can always evaluate the determinant of a square matrix. Algebraic For the following exercises, find the determinant. Technology For the following exercises, use the determinant function on a graphing utility. Real-World Applications For the following exercises, create a system of linear equations to describe the behavior.
One number is 20 less than the other. Review Exercises Systems of Linear Equations: Two Variables For the following exercises, determine whether the ordered pair is a solution to the system of equations. Systems of Linear Equations: Three Variables For the following exercises, solve the system of three equations using substitution or addition.
Systems of Nonlinear Equations and Inequalities: Two Variables For the following exercises, solve the system of nonlinear equations. Partial Fractions For the following exercises, decompose into partial fractions. Matrices and Matrix Operations For the following exercises, perform the requested operations on the given matrices. Solving Systems with Gaussian Elimination For the following exercises, write the system of linear equations from the augmented matrix.
Solving Systems with Inverses For the following exercises, find the inverse of the matrix.
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