What is the sum of the arithmetic sequence 8, 14, 20 if there are 22 terms?
To find the sum of the arithmetic sequence, we first need to determine the first term, common difference, and the total number of terms. The first term (a) is 8. The common difference (d) can be calculated by subtracting the first term from the second term: d = 14 – 8 = 6. Now that […]
What property justifies the statement: If 3x = 4 = 10, then 3x = 6?
The property that justifies this statement is known as the Transitive Property of Equality. In mathematics, the Transitive Property states that if two values are equal to a third value, then they are equal to each other. In this case, if we have 3x equal to both 4 and 10, we can assert that 4 […]
If 3y lies on the graph of y = 3x, then what is y?
To find the value of y that satisfies the equation when 3y lies on the graph of y = 3x, we can start by substituting the expression for y into the equation. First, let’s express the equation: y = 3x Since we are given that 3y lies on this graph, we can write: 3y = […]
What is the length and width of a rectangular deck if its length is 5 feet longer than its width and the area is 310 square feet?
To find the dimensions of the rectangular deck, we can set up an equation based on the information provided. Let’s denote the width of the deck as x. According to the question, the length of the deck is 5 feet longer than its width. Therefore, we can express the length as (x + 5). The […]
What is the value of x in the equation 7x + 2y = 48 when y = 3?
To find the value of x in the equation 7x + 2y = 48 when y = 3, we first substitute the value of y into the equation. Substituting y: 7x + 2(3) = 48 This simplifies to: 7x + 6 = 48 Next, we need to isolate 7x. We do this by subtracting 6 […]
What is the f of x over the g of x when f(x) = 6x^3 + 19x^2 + 16x^4 and g(x) = x^2?
To find the expression for f(x) over g(x), we need to divide the function f(x) by g(x). Given: f(x) = 6x^3 + 19x^2 + 16x^4 g(x) = x^2 We can express this as: f(x) / g(x) = (6x^3 + 19x^2 + 16x^4) / x^2 Next, we simplify this division. We can rewrite the numerator: The […]
What is the area of a rectangle with a length of 30 cm and a width of 24 cm?
The area of a rectangle can be calculated using the formula: Area = Length × Width. In this case, the length is 30 cm and the width is 24 cm. Now, substituting the values into the formula: Area = 30 cm × 24 cm This gives us: Area = 720 cm² Therefore, the area of […]
Write the Sum as a Product: Simplifying the Product 2 + 2 + 2 + 2
To simplify the sum 2 + 2 + 2 + 2, we first recognize that this is the addition of 2, four times. We can express this sum as a multiplication: Since we have four instances of the number 2, we can write this as: 2 + 2 + 2 + 2 = 4 × […]
What is the remainder when x^3 + 4x^2 + 12x + 9 is divided by x^2?
To find the remainder when the polynomial x^3 + 4x^2 + 12x + 9 is divided by x^2, we can use polynomial long division or synthetic division. However, in this case, we can also simply calculate the remainder by substituting the divisors’ roots. When we divide by x^2, the remainder will be a polynomial of […]
If x 32 42, what is the approximate positive value of x?
To solve the expression x 32 42, we need some clarification on the operation implied by the placeholder ‘x’. Assuming this is a mathematical operation, it could mean x + 32 = 42 or x * 32 = 42. Let’s evaluate both scenarios: 1. If x + 32 = 42: To find the value of […]