🧮 algebra

1. **State the problem:** Solve for $n$ in the proportion $$\frac{n}{98} = \frac{20}{35}$$. 2. **Formula and rule:** In a proportion $\frac{a}{b} = \frac{c}{d}$, the cross products

1. **State the problem:** We need to solve for $y$ in the proportion $$\frac{48}{72} = \frac{y}{69}$$. 2. **Formula used:** In proportions, if $$\frac{a}{b} = \frac{c}{d}$$, then $

1. The problem is to solve for $v$ given the equation $5 = \frac{30}{50} v$. 2. The formula used here is to isolate $v$ by multiplying both sides of the equation by the reciprocal

1. **State the problem:** Solve for $p$ in the proportion $$\frac{15}{25} = \frac{21}{p}.$$\n\n2. **Recall the property of proportions:** If $$\frac{a}{b} = \frac{c}{d},$$ then the

1. **State the problem:** Simplify the expression $$36\frac{1}{2} \times 64 - \frac{1}{3} \times 5^0$$. 2. **Recall important rules:**

1. Énoncé du problème : Comparer les deux nombres $A = \frac{n+2}{n+3}$ et $B = \frac{n+3}{n+4}$ où $n$ est un entier naturel.

1. **Problem Statement:** Prove that if $$ (1 + x)^n \geq 1 + nx $$ for $$ x > -1 $$ and $$ n \in \mathbb{N} $$, then $$ (1 + x)^{n+1} \geq 1 + (n + 1)x $$.

1. **State the problem:** Lauren starts with 25 gallons in her truck's gas tank. The pump fills the tank at 35 gallons per minute. We want to write an equation for the amount of ga

1. **State the problem:** Simplify the expression $$\frac{n+2}{n+3} - \frac{n+3}{n+4}$$. 2. **Formula and rules:** To subtract two fractions, find a common denominator and then sub

1. **Stating the problem:** We are given several polynomials and asked to factor them or determine if they are factorable using the difference of squares pattern. We also need to i

1. **State the problem:** Find the slope of the line passing through the points $(-2, -3)$ and $(1, 3)$. 2. **Formula for slope:** The slope $m$ of a line through two points $(x_1,

1. **State the problem:** Find the slope of the line passing through the points $(7, -4)$ and $(4, -6)$. 2. **Formula for slope:** The slope $m$ of a line passing through points $(

1. **State the problem:** Calculate the value of the expression $7 \times 0.007 \times \sqrt{0.0435}$.\n\n2. **Recall the formula and rules:**\n- Multiplication of decimals is stra

1. **State the problem:** Evaluate the expression $9 \circ 0.4 \sqrt{0.019}$ where $\circ$ denotes multiplication. 2. **Recall the formula:** Multiplication is associative and comm

1. **State the problem:** We are given the linear equation $y = -\frac{1}{3}x + 6$ and want to understand its properties such as slope, y-intercept, and x-intercept. 2. **Formula a

1. **Problem statement:** We are given the arithmetic sequence: $-4, -10, -16, -22, \ldots$ and need to find the explicit formula for the $n^{th}$ term $a_n$. 2. **Formula for arit

1. **State the problem:** We are given an arithmetic sequence: -16, -11, -6, -1, ... We need to find an explicit formula for the $n^{th}$ term $a_n$.

1. The problem asks us to find the explicit formula for the $n^{th}$ term $a_n$ of the arithmetic sequence: 12, 19, 26, 33, ... 2. An arithmetic sequence has a constant difference

1. **Stating the problem:** We are given the center of a circle at coordinates $(2a, a - 7)$ and a point on the circle $(11, -9)$. The diameter of the circle is $10\sqrt{2}$ units.

1. The problem is to graph the linear inequality $y > 2x - 4$. 2. The inequality represents all points $(x,y)$ where the $y$-value is strictly greater than the line $y = 2x - 4$.

1. Énoncé du problème : Comparer $a$ et $\frac{1}{a}$ pour $a > 0$. 2. Justification de l'identité : Montrons que