🧮 algebra

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

1. Problem: Find the equation of the line passing through point A(2,4) and parallel to the line $y=5x-3$. 2. Formula: Lines parallel to $y=mx+b$ have the same slope $m$. Here, slop

1. **State the problem:** Find the product of $$\left(\frac{3}{5}x - \frac{y}{2}\right) \left(\frac{5}{3}x + 6y\right)$$ and verify the result for $$x = -1$$ and $$y = 2$$. 2. **Us

1. **State the problem:** Solve for $n$ in the equation $$\frac{5}{3} = \frac{5}{n+2} - \frac{n}{6n}.$$\n\n2. **Rewrite the equation:** Note that $\frac{n}{6n} = \frac{1}{6}$ for $

1. **State the problem:** We need to find the equation of a line with slope $-\frac{3}{5}$ that passes through the point $(-3,3)$. The equation should be in slope-intercept form $y

1. **State the problem:** We need to find the equation of a line with slope $m=3$ that passes through the point $(-2,-15)$ in slope-intercept form $y=mx+b$. 2. **Recall the slope-i

1. The problem asks us to find the equation of a line in slope-intercept form given the slope and a point on the line. 2. The slope-intercept form of a line is given by the formula

1. **State the problem:** We need to find the equation of a line with slope $\frac{3}{5}$ that passes through the point $(11,6)$ in slope-intercept form $y=mx+b$. 2. **Recall the s

1. The problem asks for the equation of a line with slope 0 passing through the point $(5,-16)$. 2. Recall the slope-intercept form of a line is given by $$y = mx + b$$ where $m$ i

1. **Problem Statement:** Prove by mathematical induction that for all integers $n \geq 2$,

1. **Problem statement:** Prove by induction that $$\sum_{r=1}^n r = \frac{n(n+1)}{2}$$ for all natural numbers $n$. 2. **Base case:** For $n=1$, the left side is $1$ and the right

1. **State the problem:** We need to find the equation of a line with slope $-4$ passing through the point $(2,-15)$ in slope-intercept form $y=mx+b$. 2. **Recall the slope-interce

1. Bài toán yêu cầu giải phương trình $$x^2 - 3x + (x - 3) = 0$$. 2. Ta bắt đầu bằng cách rút gọn biểu thức trong dấu ngoặc:

1. Write as a single power **a.** $3^2 \times 3 = 3^{2+1} = 3^3$

1. **State the problem:** Write $a\ 3^2 \times 3$, $b\ 7 \times 7^3$, $c\ 12 \times 12^5$, and $d\ 15^4 \times 15$ as a single power. 2. **Recall the rule for multiplying powers wi