MIT

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question:Find the inverse of the matrix [begin{pmatrix} X & 18 -6 & -12 end{pmatrix}.]If the inverse does not exist, then enter the zero matrix. The answer is 0. What is the value of unknown variable X?

answer:To find the inverse of a matrix, we need to determine the value of x, which represents the unknown variable in the given matrix. To find the inverse of a 2x2 matrix, we can use the formula: [ begin{pmatrix} a & b c & d end{pmatrix}^{-1} = frac{1}{ad - bc} begin{pmatrix} d & -b -c & a end{pmatrix}.] In our case, we have the matrix: [ begin{pmatrix} X & 18 -6 & -12 end{pmatrix}.] To find the inverse, we need to calculate the determinant, which is given by: [ ad - bc.] In our case, the determinant is: [ X(-12) - (18)(-6) = -12X + 108.] If the determinant is zero, then the inverse does not exist. Therefore, we need to find the value of x for which the determinant is not zero. Setting the determinant equal to zero: [ -12X + 108 = 0.] To solve for x, we isolate x: [ -12X = -108.] [ X = frac{-108}{-12}.] Simplifying, we have: [ X = 9.] Therefore, the value of the unknown variable x is 9. The answer is: 9

question:Julio has x bottles of orange soda and seven bottles of grape soda in his fridge. His friend Mateo has a bottle of orange soda and 3 bottles of grape soda in his fridge. If the amount of beverage in each bottle is 2 liters, how many more liters of soda does Julio have? If we know the answer to the above question is 14, what is the value of unknown variable x?

answer:Julio has x bottles of orange soda, each containing 2 liters, so he has a total of 2x liters of orange soda. He also has 7 bottles of grape soda, each containing 2 liters, so he has a total of 7 * 2 = 14 liters of grape soda. Mateo has 1 bottle of orange soda, containing 2 liters, and 3 bottles of grape soda, each containing 2 liters, so he has a total of 1 * 2 + 3 * 2 = 8 liters of soda. To find out how many more liters of soda Julio has, we subtract the total amount of soda Mateo has from the total amount of soda Julio has: 2x + 14 - 8 = 2x + 6. We are given that the difference is 14 liters, so we can write: 2x + 6 = 14. Solving for x, we get: x = 4. The value of x is 4. 4 The answer is: 4

question:Thomas is training at the gym to prepare for a competition. He trained for 5 hours every day for a month (30 days). If he continues to train for the next 12 days, how many hours will he spend on training in total?

answer:Thomas trained for 5 hours every day for a month, so he trained for 5 hours x 30 days = 150 hours in total. If he continues to train for the next 12 days, he will spend an additional 5 hours x 12 days = 60 hours on training. Therefore, in total, Thomas will spend 150 hours + 60 hours = 210 hours on training. 210 The answer is: 210

question:Emery looks through the drawer where her mom keeps the cookie cutters. There are 6 cookie cutters shaped like triangles, x square ones and 2 hexagons. There are 46 sides on the cookie cutters in total. What is the value of unknown variable x?

answer:To solve this problem, we need to determine the value of x, which represents the number of square cookie cutters that Emery's mom has. Let's break down the information given: Number of triangle cookie cutters: 6 Number of square cookie cutters: x Number of hexagon cookie cutters: 2 Total number of sides on the cookie cutters: 46 We can set up the equation as follows: Number of triangle cookie cutters * Number of sides per triangle + Number of square cookie cutters * Number of sides per square + Number of hexagon cookie cutters * Number of sides per hexagon = Total number of sides on the cookie cutters 6 * 3 + x * 4 + 2 * 6 = 46 Let's simplify and solve for x: 18 + 4x + 12 = 46 To isolate x, we subtract 30 from both sides of the equation: 18 + 4x + 12 - 30 = 46 - 30 4x = 16 To solve for x, we divide both sides of the equation by 4: x = 16 / 4 x = 4 The value of x is 4. 4 The answer is: 4