How to Solve the Linear Equations 2x + y = 5 and 3x + 2y = 4?
To solve the system of linear equations given by: 1. 2x + y = 5 2. 3x + 2y = 4 we can use the substitution or elimination method. Here, we’ll use the substitution method. First, let’s solve the first equation for y: y = 5 – 2x Now that we have y in terms […]
Find the Derivative of the Function Using the First Principle of Derivative
To find the derivative of a function using the first principle of derivatives, we start with the definition of the derivative: Given a function f(x), the derivative f'(x) is defined as: f'(x) = lim (h → 0) [(f(x + h) – f(x)) / h] This limit represents the slope of the tangent to the function […]
How to Find the Area of a Triangle with Base of 15m and Altitude of 12m?
To find the area of a triangle, you can use the formula: Area = 1/2 × base × height In this case, the base of the triangle is 15 meters and the height (altitude) is 12 meters. Plugging these values into the formula gives: Area = 1/2 × 15m × 12m Calculating this step by […]
What is the greatest common factor of 60x^4y^7, 45x^5y^5, and 75x^3y?
The greatest common factor (GCF) is the largest expression that divides each term in a set of terms without leaving a remainder. To determine the GCF of the expressions 60x4y7, 45x5y5, and 75x3y, we need to find the GCF for both the numerical coefficients and the variable parts separately. 1. **Finding the GCF of the […]
What are the coordinates of the vertex of the function f(x) = x² + 10x + 3?
To find the coordinates of the vertex of the quadratic function f(x) = x² + 10x + 3, we can use the vertex formula. The x-coordinate of the vertex for any quadratic in the form f(x) = ax² + bx + c can be found using the formula: x = -b / (2a) In our […]
The Number of Integers Between 100 and 500 that are Multiples of 11
To find the integers between 100 and 500 that are multiples of 11, we first need to identify the smallest and largest multiples of 11 within this range. The smallest multiple of 11 that is greater than or equal to 100 can be found by dividing 100 by 11 and rounding up to the nearest […]
What is the age of A if A is elder to B by 2 years, and F is twice as old as A, while B is twice as old as his sister S, given that the age difference between Father F and Sister S is 40 years?
Let the age of A be a years. According to the question: A is elder to B by 2 years, so B’s age is b = a – 2. Father F is twice as old as A, thus f = 2a. B is twice as old as his sister S, leading to s = b […]
Find the Maximum Volume of a Rectangular Box Inscribed in a Sphere of Radius r
To find the maximum volume of a rectangular box inscribed in a sphere of radius r, we can start by considering the relationship between the box and the sphere. Let the dimensions of the box be x, y, and z. Since the box is inscribed in the sphere, the corners of the box touch the […]
If an equation of the linear function is y = mx + b, then what is m?
In the equation of a linear function, y = mx + b, the letter m represents the slope of the line. The slope is a measure of how steep the line is and indicates the rate of change of the y-value with respect to the x-value. More specifically, the slope m quantifies how much y […]
What is the product of 5r, 4r², 6r, and 4?
To find the product of the terms 5r, 4r², 6r, and 4, we will multiply these expressions together step by step. First, let’s group the numerical coefficients together and the variable parts together: 1. Numerical coefficients: 5, 4, 6, and 4. 2. Variables: r, r², and r. Now we’ll multiply the numerical coefficients: 5 * […]