🧮 algebra

1. **State the problem:** Kirstie charges $18 for the first hour and $14 for each additional hour. The total charge $C$ is given by the formula $$C = 18 + 14(h - 1)$$ where $h$ is

1. **State the problem:** Factorise the quadratic expression $8x^2 + 22x + 15$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that m

1. **Problem:** Prove algebraically that for all positive integers $n$, the expression $n^3 + 3n^2 - 2n$ is always even. 2. **Formula and rules:** An integer is even if it is divis

1. **Problem statement:** Prove algebraically that for all positive integers $n$, the expression $n^3 + 3n^2 - 2n$ is always even. 2. **Formula and rules:** An integer is even if i

1. **Problem Statement:** Determine if the given relations are functions by analyzing their domain, range, and the vertical line test. 2. **Key Concept:** A relation is a function

1. **State the problem:** Simplify the expression $$\frac{2x^{-3}y^{4}}{4x^{-1}y^{-2}}$$ and write it using only positive indices. 2. **Recall the rules:**

1. **State the problem:** We are given the system of equations: $$y = -x + 2$$

1. The problem is to find the formula for the geometric sequence given points (1, 9) and (2, \frac{27}{5}). 2. Recall the formula for the $n$-th term of a geometric sequence:

1. **State the problem:** We are given three points $(1, -5)$, $(-2, -6)$, and $(-5, -7)$ and asked to find the formula for the arithmetic sequence $a_n$ that fits these points. 2.

1. **State the problem:** Calculate the value of the expression $2491.67 - 711.67(1) - 1511.67(0) + 326.67(0)$. 2. **Apply multiplication:** Multiply each number inside the parenth

1. **State the problem:** Solve the equation $$d^2 - 24 = 0$$ for $d$. 2. **Formula and rules:** This is a quadratic equation in the form $$d^2 = 24$$. To solve for $d$, we take th

1. **State the problem:** We are given two linear equations: $$2x + y = 7$$

1. **State the problem:** Convert 3°C to Fahrenheit using the formula $$C = \frac{5}{9}(F - 32)$$. 2. **Formula and explanation:** The formula relates Celsius (C) and Fahrenheit (F

1. The problem is to simplify an expression, but since no specific expression was provided, I cannot simplify anything. 2. Please provide the expression you want to simplify so I c

1. **Problem statement:** Find a quadratic equation whose roots are $\alpha + \frac{1}{\beta}$ and $\beta + \frac{1}{\alpha}$, given that $\alpha$ and $\beta$ are roots of the equa

1. The problem asks to find the equation of the line passing through the points (-7, 3) and (6, -6). 2. The formula for the slope $m$ of a line passing through points $(x_1, y_1)$

1. The problem asks to graph the function $f(x) = x$ after it has been vertically stretched by a factor of 2 and vertically shifted down by 3 units. 2. The original function is $f(

1. **State the problem:** We need to find the equation of the line passing through the points $(-7,6)$ and $(7,-6)$. 2. **Formula used:** The equation of a line in slope-intercept

1. **State the problem:** Solve for $x$ in the equation $$\frac{2x - 3}{3} + \frac{3x + 4}{5} = 15.$$\n\n2. **Identify the formula and rules:** To solve this equation, we need to c

1. **State the problem:** Simplify the expression $$5^7 \div (5^2 \times 5^3)$$. 2. **Recall the laws of exponents:**

1. **Stating the problem:** We need to find a number such that increasing it by 14 and then doubling the result equals decreasing the number by 27 and tripling that result. 2. **Cr