To solve the system of equations by graphing, we will analyze both equations separately and then identify their point of intersection, which represents the solution to the system. Step 1: Rearrange the equations The given equations are: Equation 1: y = 3x + 1 Equation 2: y = x + 7 Both equations are already … Read more
To find the general solution of the second-order differential equation: y” + 8y’ + 17y = 0 we first rewrite it in standard form: 1. **Characteristic Equation**: We start by forming the characteristic equation associated with the differential equation. This is done by substituting y = e^{rt}, where r is a constant. We differentiate this … Read more
To determine the probability of guessing all answers correctly on a multiple-choice test with 5 questions, each having 5 possible answers, we can follow these steps: 1. **Understanding the Basics**: For each question, there are 5 possible answers. If you’re guessing, the chance of selecting the correct answer for one question is 1 out of … Read more
To find the value of x in the equation 10x + 2 – 5x – 8 = 0, we first need to simplify the equation. Follow these steps: Combine like terms: First, let’s group the x terms together and the constant terms together: 10x – 5x 2 – 8 This gives us: 5x – 6 … Read more
Understanding the Difference of Squares The difference of squares is a common algebraic factorization technique that takes the form of a² – b² = (a – b)(a + b). We’ll analyze the expression provided and identify notable patterns that may indicate a difference of squares. Breaking Down the Expression The given expression can be stated … Read more
To find the remainder when the polynomial x4 + 36 is divided by x2 + 8, we can use polynomial long division or synthetic division. However, for many polynomial operations, a more straightforward approach is to express the dividend in a suitable form. 1. **Expressing the Dividend:** Start with: x4 + 36 2. **Dividing Terms**: … Read more
The original price of the pants is $3500. To calculate the discount amount, you need to first find 20% of the original price. Here’s how you can do it step by step: Convert the percentage to a decimal by dividing by 100: 20% = 20/100 = 0.20. Multiply the original price by the decimal: $3500 … Read more
To find the value of the discriminant for the quadratic equation 3x² + 2x, we first need to identify the coefficients in the standard form of a quadratic equation, which is generally expressed as ax² + bx + c = 0. Here, we can rewrite the equation: 3x² + 2x = 0 can be interpreted … Read more
To understand how many cubic feet are in one unit of sand, we first need to clarify what ‘unit of sand’ means, as this terminology can vary based on context and location. Often, one unit of sand can refer to a cubic yard, especially in construction and landscaping. If we take a cubic yard as … Read more
The least common multiple (LCM) of two numbers is the smallest multiple that is evenly divisible by both numbers. To find the LCM of the denominators 16 and 58, we can use the following steps: Prime Factorization: First, we will find the prime factorization of both numbers. For 16: 16 = 2 × 2 × … Read more