🧮 algebra

1. **Problem statement:** Solve for $x$ in the equation $$x - \frac{1}{x} = \sqrt{x} + \frac{1}{\sqrt{x}}$$ and then evaluate $$x^y + x^{-y}$$. 2. **Step 1: Simplify the given equa

1. **Problem:** Find the solution set of the system of equations: $$\begin{cases} x - y = 1 \\ 5x^2 + 2y^2 = 53 \end{cases}$$

1. **State the problem:** We have the equation $$2x^2 - 3xy - y^2 + x - 5y - 3 = 0$$ and want to translate the origin to a point $ (h,k) $ so that the new equation has no first deg

1. The problem is to compare the value of $4\sqrt{2}$ with other numbers or expressions. 2. Recall that $\sqrt{2}$ is approximately equal to 1.414.

1. The problem asks for the range of the function defined by the equation $$y = x^2 - 3x + 5$$. 2. This is a quadratic function in the form $$y = ax^2 + bx + c$$ where $$a=1$$, $$b

1. The problem asks for the equation of a line that passes through the point $(0,3)$ and is perpendicular to the line given by $$y = -x + 1.$$\n\n2. The slope of the given line is

1. **State the problem:** Evaluate the expression $$5 + \frac{3}{5 + \frac{3}{5 - 3}}$$. 2. **Understand the structure:** This is a nested fraction (a complex fraction) where the d

1. **State the problem:** Calculate the value of the expression $$\frac{1}{\frac{0.3}{0.33} + \frac{0.5}{0.55} + \frac{0.7}{0.77}}$$ and select the correct answer from the options

1. The problem is to evaluate the expressions $2\left(\frac{7}{3}\right)$ and $\frac{1}{2}\left(\frac{8}{3}\right)$. 2. Recall the rule for multiplying a whole number or fraction b

1. The problem is to simplify the expression $3,21$. 2. It appears there might be a typo or formatting issue since $3,21$ is not a standard mathematical expression. Assuming you me

1. **State the problem:** Factorise the expression $a(b-c)^2 - d(c-b)$. 2. **Rewrite the expression:** Notice that $c-b = -(b-c)$, so replace $d(c-b)$ with $-d(b-c)$. The expressio

1. **State the problem:** Solve the quadratic equation $$x^2 - 3x - 6 = 0$$. 2. **Formula used:** The quadratic formula for solving $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqr

1. **State the problem:** Solve the quadratic equation $2x^2 - 3x - 6 = 0$ by completing the square. 2. **Rewrite the equation:** Divide all terms by 2 to make the coefficient of $

1. The problem is to find the missing value of $x$ in the ordered pair $(x, -6)$ that satisfies the equation $4x + y = 10$. 2. Substitute $y = -6$ into the equation: $4x + (-6) = 1

1. Problem: Compute the matrix operations for given matrices A, B, and C. Matrices:

1. **State the problem:** Find the slope of the line given points on the graph. 2. **Recall the slope formula:** The slope $m$ of a line passing through two points $(x_1, y_1)$ and

1. The problem asks for the slope of the line given by the equation $$4x - 1 = 3y + 5$$. 2. To find the slope, we need to rewrite the equation in slope-intercept form $$y = mx + b$

1. The problem is to find the missing value of $y$ when $x = -1$ in the equation $y - 3 = 2(x + 1)$. 2. The formula given is a linear equation in point-slope form. We will substitu

1. The problem is to verify or understand the formula for the sum of squares of the first $n$ natural numbers: $$12 + 22 + 32 + \cdots + n^2 = 6n(n+1)(2n+1)$$ 2. The correct formul

1. The problem is to graph the line given by the slope-intercept form equation $$y = -\frac{1}{3}x + 5$$. 2. The slope-intercept form of a line is $$y = mx + b$$ where $m$ is the s

1. **Problem 1: Transformation of figure ABCD under ERS** Given points: