🧮 algebra

1. **State the problem**: Isaac approximated $20 \times 300 = 6000$ by rounding each number to 1 significant figure. We need to find the smallest possible exact product of the orig

1. **State the problem:** There are vans, lorries, and motorbikes in the ratio 2 : 5 : 9. 2. We know that the number of vans corresponds to 2 parts in the ratio.

1. Stating the problem: Solve the system of equations: $$x + y = 5$$

1. Problem: Find the slope of the line through points (1, 3) and (4, 9). Step 1: Use the slope formula $$m = \frac{y_2 - y_1}{x_2 - x_1}$$.

1. Stating the problem: We are given the matrix $$\begin{pmatrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{pmatrix}$$

1. The question asks whether a groupoid and a binary operation are the same and if they both come under algebraic structures. 2. A \textbf{binary operation} on a set $S$ is a funct

1. We are given a cubic equation $$x^3 + ax^2 + bx + c = 0$$ with roots $$\alpha, \beta, \gamma$$ and the system: $$\alpha u + \beta v + \gamma w = 0,$$

1. **Problem statement:** We need to find the values of $a$ and $b$ such that the coefficients of $x^3$ and $x^4$ in the expansion of $\left(1 + ax + bx^2\right)(1 - 2x)^{18}$ are

1. The problem is to simplify the expression **-2(3m - n + 4)**. 2. Distribute the **-2** to each term inside the parentheses:

1. **State the problem:** Find the value of the sum $$\sum_{r=1}^n {^nC_r} \alpha_r$$ where

1. We are given four pairs of simultaneous equations to solve for $x$ and $y$: (1) $x + y + xy = 5$, $x^2 + y^2 = 1$

1. Calculate the value of $r$ given $r = 2a - \frac{\sqrt{b}}{c}$, $a = 0.975$, $b = 4.41$, $c = 35$. Step 1: Calculate $\sqrt{b} = \sqrt{4.41}$.

1. The problem is to simplify the expression $3(a+b-2)$. 2. Apply the distributive property which states $c(x+y) = cx + cy$. Here, multiply each term inside the parentheses by 3:

1. The problem is to simplify the expression $\frac{\sqrt{5-7}}{\sqrt{5}+\sqrt{7}}$. 2. First, simplify inside the square root in the numerator: $5 - 7 = -2$.

1. The problem asks to find the greatest common divisor (GCD) for two pairs of integers involving the variable $n$. 2. For part (a), find $\gcd(-7, \ 2n + 3)$.

1. The problem asks for the maximum number of people who live in Birmingham given that the population is rounded to the nearest 1000 as 1 020 000. 2. When rounding to the nearest 1

1. The problem is to solve the inequality $$-| -6v - 5| - 7 < -19$$ for the variable $v$. 2. First, isolate the absolute value term. Add 7 to both sides:

1. We are asked to solve the inequality $$4|4p - 5| - 6 \leq 10$$ for $p$. 2. Start by isolating the absolute value term:

1. **State the problem:** Solve the inequality $$-3|{-t}| - 3 \leq -16$$ for the variable $$t$$. 2. **Simplify the absolute value expression:** Note that $$|{-t}| = |t|$$ because a

1. We are given the inequality $$\frac{2 - x}{x - 2} > \frac{x}{x}$$ and the domain restriction $$1 \leq x < 3$$. 2. Notice that $$\frac{2 - x}{x - 2} = \frac{-(x - 2)}{x - 2} = -1

1. **State the problem**: Solve the inequality $$4 - 8|w - 7| \geq -20$$ for the variable $$w$$ and express the solution as a compound inequality. 2. **Isolate the absolute value t