🧮 algebra

1. **Determine whether the following statements are functions of $x$ or not.** (i). $y = x^2 + 1$ for each $x \in \mathbb{R}$.

1. **State the problem:** Find the value of $?$ in the equation $$45.24 \times 89.92 + 28.39 \times 90.23 = ? - 3.44.$$ 2. **Rewrite the equation:** Add $3.44$ to both sides to iso

1. **State the problem:** Find the value of $\sqrt{24}$.\n\n2. **Recall the formula and rules:** The square root of a number $x$ is a value that, when multiplied by itself, gives $

1. The problem is to simplify the expression \(\sqrt{x}24\). 2. We interpret this as \(24\sqrt{x}\), which means 24 times the square root of \(x\).

1. The problem is to sketch the graph of the function $$y = -2(x + 1)^2 - 3$$ by transforming the graph of $$y = x^2$$. 2. The base function is $$y = x^2$$, which is a parabola ope

1. **State the problem:** We are given the sum of the first 10 terms ($S_{10}$) of an arithmetic progression (AP) as 460 and the sum of the next 10 terms ($S_{20} - S_{10}$) as 126

1. The problem is to analyze the system of 10 linear equations in variables $x$, $y$, and $z$: $$\begin{cases}

1. **Stating the problem:** We are given the sequence of numbers 4, 6, 10 and asked to find the pattern. 2. **Observing the sequence:** The numbers are 4, 6, 10.

1. The problem asks to find the pattern of a sequence of numbers, but no numbers were provided. 2. To find a pattern, we typically look for relationships such as arithmetic progres

1. **Stating the problem:** We are given the sequence 2, 8, 18, 32, 50 and need to find the formula for the nth term. 2. **Look for a pattern:** Let's examine the differences betwe

1. **Stating the problem:** We are given the sequence 2, 8, 18, 32, 50 and asked to find the pattern of the terms. 2. **Look for a pattern:** Let's denote the $n$th term as $a_n$.

1. **Problem statement:** Solve the equation $$\sqrt{3} - x = x\sqrt{3} + x.$$\n\n2. **Rewrite the equation:** Move all terms to one side to isolate $x$:\n$$\sqrt{3} - x - x\sqrt{3

1. **State the problem:** Given that $\left(x+\frac{1}{x}\right)^2=40$, find the value of $x^2+\frac{1}{x^2}$.\n\n2. **Recall the formula:** We know that $\left(a+b\right)^2 = a^2

1. **State the problem:** We are given the system:

1. **Stating the problem:** Given the equation $$\frac{x}{x^2 + 3x - 1} = \frac{1}{4}$$, find the value of $x$. 2. **Rewrite the equation:** Multiply both sides by the denominator

1. **State the problem:** Write the constants in the expression $8x^2 + 2y + 5$. 2. **Identify constants:** Constants are terms that do not change with variables. In this expressio

1. **Problem:** Find the equation of the straight line which passes through the point (2, -3) and is parallel to the x-axis. 2. **Concept:** A line parallel to the x-axis has a con

1. The problem is to solve for $x$ in the equation $2x + 3 = 7$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. The problem asks to find the equations of lines symmetric to the given lines with respect to the line $y=1$. 2. The line of symmetry is $y=1$. For any point $(x,y)$ on the origi

1. **State the problem:** We are given the function $F(x) = 1 + \frac{1}{x^2} + 1$ and need to simplify it. 2. **Combine like terms:** Notice that $1 + 1 = 2$, so the function beco

1. **State the problem:** We need to find the rate of flow of water in litres per second given that a water tap fills a 500 litres tank in 10 minutes. 2. **Formula used:** Rate of