🧮 algebra

1. **State the problem:** Solve the system of equations using the elimination method: $$\begin{cases}-x - 5y - 5z = 2\\4x - 5y + 4z = 19\\x + 5y - z = -20\end{cases}$$

1. State the problem: Solve the equation $3x + 5 = 7$ for $x$. 2. Subtract 5 from both sides to isolate the term with $x$:

1. Problem: For each sequence, find the next three terms and the rule for the terms. 2. Sequence a: $1, 2, 4, 8, 16, 32, \dots$

1. **Stating the problem:** We have two products to analyze and compare: - Product A: $(n+1)(n+2)(n+3)\ldots(n+2s-1)(n+2s)$

1. Stating the problem: Solve the equation $$|\frac{3 - 2x}{x + 6}| = 4$$ for $x$. 2. Breaking down the absolute value equation, we get two cases:

1. The problem states that green and yellow dyes are mixed in a ratio of 5:6 to make 44 litres of lime-coloured dye. 2. We want to find how much green and yellow dye there are in t

1. **Problem:** Find the domain of the function $$y=\frac{x+1}{\sqrt{x-1}}$$. - The expression inside the square root must be positive because the denominator cannot be zero or neg

1. The user requests to exchange the minus sign (-) for an equal sign (=). 2. This implies changing an expression like $a - b$ to $a = b$.

1. The problem asks us to explain the function $f(a+3) = (a+3)^2$. 2. This function represents the square of the expression $a+3$.

1. Stated problem: Solve the equations \(2x - 3 \cdot 54 + 2 = 1000\) and \(\sqrt{x - 27} = 6\).\n\n2. For the first equation, simplify and solve for \(x\):\nCalculate \(3 \cdot 54

1. The problem is to solve the quadratic equation $x^2 - x - 1 = 0$ for $x$. 2. We use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = 1$, $b = -1$, an

1. **Rationalise the following:** a. $\frac{1 + \sqrt{3}}{2 + \sqrt{3}}$

1. The problem asks to find the result of $50_60$ which is normally interpreted as 50 minus 60. 2. Subtracting 60 from 50: $50 - 60 = -10$.

1. Stated problem: Simplify the expression $$\frac{7ab^3}{10x^2 y^3} : \frac{21a^2 b^2}{15 x^3 y^2}$$. 2. Recall that dividing by a fraction is the same as multiplying by its recip

1. The problem is to solve and explain the expression 50:60. 2. The colon ":" typically represents a ratio in mathematics.

1. **State the problem:** Find the values of $x$ and $y$ such that $$(3x - y) + (x - 3y)i = 5 - i.$$\n\n2. **Identify real and imaginary parts:** For two complex numbers to be equa

1. The problem gives us two exchange rates for pounds (£) to Australian dollars ($) from two travel agents, Aussie Holidays and Kangaroo Travel, both starting at the origin (0,0).

1. Problem: Find the magnitude and slope of vectors $\mathbf{A}$ and $\mathbf{B}$ where $\mathbf{A}=6+ j8$ and $\mathbf{B}=3- j4$. 2. Magnitude of a vector $\mathbf{v}=x+jy$ is giv

1. **State the problem:** We are given two equations: $$a x_1^2 + b x_1 y_1 + c y_1^2 + d = 0$$

1. **State the problem:** Simplify the expression $$\sqrt{x} - 20\sqrt{x^{10}} + 4\sqrt{x^2} - 6\sqrt{x^3} + 8\sqrt{x^4}$$ in radical form.\n\n2. **Rewrite each term using exponent

1. Stating the problem: We need to find the Least Common Multiple (LCM) and Highest Common Factor (HCF, also called GCD) of given numbers. 2. Explanation: The HCF of two or more nu