A ladder 10 ft long rests against a vertical wall. Let theta be the angle between the top of the ladder and the wall and let x be the distance.
In this scenario, we have a classic right triangle formed by the ladder, the wall, and the ground. The ladder itself serves as the hypotenuse, which is 10 feet long. The height at which the ladder touches the wall can be represented as the opposite side of the triangle, and the horizontal distance from the […]
How do you simplify the expression 2n² + 4n + 44n + 5 using the horizontal method?
To simplify the expression 2n² + 4n + 44n + 5 using the horizontal method, we will first combine like terms. 1. Start by identifying the like terms in the expression. In this case, we have: 2n² (the quadratic term) 4n and 44n (the linear terms) 5 (the constant term) 2. Combine the linear terms: […]
How to Solve the System of Equations: x + 3y = 1 and 2x + 2y = 6
To solve the system of equations given by: Equation 1: x + 3y = 1 Equation 2: 2x + 2y = 6 We can use the method of substitution or elimination. Here, I’ll demonstrate the elimination method. First, let’s rearrange both equations for easier manipulation: From Equation 1, we can express x in terms of […]
How to Solve the Equation 5x² + 25x + 0?
To solve the equation 5x² + 25x + 0, we can start by factoring out the common term from each component of the equation. First, notice that each term shares a factor of 5x
Solve the equations by factorisation: 6x² – 150 = 0
To solve the equation 6x² – 150 = 0 by factorisation, we first want to simplify the equation. Start by moving the constant term to the other side: 6x² = 150 Next, divide both sides by 6: x² = 25 Now, we can take the square root of both sides: x = ±5 Thus, the […]
The sum of two numbers is 28 and their difference is 4. What are the two numbers?
To find the two numbers, let’s denote them as x and y. We know that: The sum of the two numbers: x + y = 28 The difference between the two numbers: x – y = 4 Now, we can solve these two equations step by step. First, let’s solve for x in terms of […]
Solve the Following System of Equations
To solve the system of equations: 2x + 3y + z = 1 3x + y + 2z = 12 x + 2y + 3 = 5 We will use the method of substitution or elimination. Let’s first rewrite the equations in a more standard format: Equation 1: 2x + 3y + z = 1 […]
Find Two Unit Vectors That Make an Angle of 60° with Vector (3, 4)
To find two unit vectors that make an angle of 60° with the vector v = (3, 4), we will use the concept of the dot product and unit vectors. First, let’s calculate the magnitude of vector v: Magnitude of v = √(3² + 4²) = √(9 + 16) = √25 = 5. Next, we […]
If f(x) = 16x + 30 and g(x) = 14x + 6, for which value of x does f(g(x)) = 0?
To solve for the value of x such that f(g(x)) = 0, we first need to find g(x) and then substitute it into f(x). Given: f(x) = 16x + 30 g(x) = 14x + 6 Now, we can find f(g(x)). First, we need to substitute g(x) into f(x): f(g(x)) = f(14x + 6) Now, replace […]
How do you solve the differential equation xy² + x²y = 0?
To solve the differential equation xy² + x²y = 0, we start by factoring the left-hand side. This gives us: y(xy + x²) = 0 This equation consists of two factors that can each equal zero. Thus, we can break it down into two separate cases: Case 1: y = 0 In this case, the […]