🧮 algebra

1. **State the problem:** Solve the equation $$\frac{4x - 1}{2} = x + 7$$ for $x$. 2. **Formula and rules:** To solve linear equations, we aim to isolate $x$ on one side by perform

1. The problem asks for the first three terms of the sequence defined by the nth term formula: $$a_n = 3 \times 2^{n-2}$$. 2. To find each term, substitute $n=1$, $n=2$, and $n=3$

1. **State the problem:** Solve the equation $$e^{-\ln x} = 20$$ for $x$. 2. **Recall the properties:** The natural logarithm and exponential functions are inverses, so $$e^{\ln a}

1. **State the problem:** Find the gradient (slope) of the line passing through the points $(-3,0)$ and $(0,-6)$.\n\n2. **Formula:** The gradient $m$ of a line through points $(x_1

1. The problem involves understanding the relationship between cost and mass of sweets for two packaging types: Box and Bag. 2. The graph shows cost (pence) on the vertical axis an

1. The problem involves interpreting expressions related to Fibonacci numbers and matching them to verbal descriptions. 2. Recall the Fibonacci sequence $F_n$ is defined as $F_0=0$

1. The problem asks to simplify the expression $$y = 2 \ln(e^x)$$. 2. Recall the logarithm rule: $$\ln(a^b) = b \ln(a)$$.

1. The problem involves understanding the relationship between the total cost and the number of packets of candies. 2. We are given a total cost of Rs. 12 for 8 packets of candies.

1. **Stating the problem:** We need to calculate percentages using the formula $$\frac{\text{part}}{\text{whole}} \times 100$$ for each given fraction. 2. **Formula explanation:**

1. The problem is to find the roots of the equation $y = x^2$ from the graph. 2. Roots of an equation are the values of $x$ for which $y = 0$.

1. **State the problem:** Simplify the expression $-e^{-\ln 21}$. 2. **Recall the properties of logarithms and exponents:** For any positive number $a$ and any real number $x$, $e^

1. The problem is to understand and analyze the linear equation $\frac{1}{4}x + 7 = y$. 2. This is a linear function in the form $y = mx + b$, where $m$ is the slope and $b$ is the

1. The problem is to analyze the function $y = x^2$ and understand its graph. 2. The formula used is $y = x^2$, which represents a parabola opening upwards.

1. **State the problem:** Simplify the expression $$\frac{6x^2 - 7x + 2}{4x^2 - 1}$$. 2. **Recall the formula and rules:** To simplify a rational expression, factor both numerator

1. The problem is to write a mathematical expression or solution "in full," which typically means expanding or expressing it completely without abbreviations or shortcuts. 2. Since

1. **State the problem:** Solve the equation $$2^x - 2^{-x} = 1$$ for $x$. 2. **Rewrite the equation:** Recall that $2^{-x} = \frac{1}{2^x}$. So the equation becomes:

1. **State the problem:** Solve the equation $$3^x = 2 + 3^{-x}$$ correct to 3 significant figures. 2. **Rewrite the equation:** Let $$u = 3^x$$. Then $$3^{-x} = \frac{1}{3^x} = \f

1. Das Problem lautet: Löse die Gleichung $2x + 3 = 7$.\n\n2. Die allgemeine Formel zur Lösung linearer Gleichungen der Form $ax + b = c$ ist: $x = \frac{c - b}{a}$.\n\n3. Wende di

1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases} \frac{(x+2)(x-2)}{x-2}, & x \neq 2 \\ 1, & x = 2 \end{cases}$$

1. The problem is to find the slope of the line through each pair of points given. 2. The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

1. **State the problem:** Ani is 3 years younger than Dion. The sum of their ages is at most 50 years. We need to find the maximum possible age of Ani 5 years ago. 2. **Define vari