🧮 algebra

1. **Problem:** Find the points of intersection of the curve $2x^2 - y + 19 = 0$ and the line $y + 11x = 4$. 2. **Step 1: Express $y$ from the line equation.**

1. **State the problem:** We have a crew where 12 men make up 60% of the total crew. We need to find the total number of people in the crew. 2. **Formula and explanation:** Let the

1. The problem is to solve the equation involving two special operators \(\oplus\) and \(\boxplus\) and an unknown \(\square\): $$5 \oplus 1 \boxplus \square = 3$$

1. **State the problem:** We need to sketch the graph of the function $$y=\frac{x^2}{x^2-4}$$. 2. **Identify the domain:** The denominator cannot be zero, so solve $$x^2-4=0$$ whic

1. **State the problem:** Simplify the expression $\frac{9}{4} - 1$. 2. **Recall the rule:** To subtract a whole number from a fraction, convert the whole number to a fraction with

1. **State the problem:** Simplify the expression $\frac{9}{4} - \frac{3}{4} \times \frac{10}{3} \div \frac{5}{2}$. 2. **Recall the order of operations:** Multiplication and divisi

1. **State the problem:** Solve the equation $|x - a| + |x + a| = 4$ for $a$ given a fixed $x$. 2. **Recall the definition of absolute value:**

1. **State the problem:** Solve the equation $|x - a| + |x + a| = 4$ for $x$. 2. **Recall the definition of absolute value:**

1. **Énoncé du problème :** Montrer que les vecteurs $a = (1, -2, 0)$, $b = (-1, 2, -3)$ et $e = (-2, 2, -1)$ forment une base de $\mathbb{R}^3$.

1. **Problem statement:** Factorise each of the following expressions completely. 2. **Recall the factoring rule:** To factorise an expression, find the greatest common factor (GCF

1. **Problem (a): Simplify \(\frac{x^2 + 7x}{x^2 - 49}\)** 2. First, factor both numerator and denominator:

1. **பிரச்சினையை விளக்குதல்:** $r \in \mathbb{Z}^+$ என்றால், $U_r = \frac{1}{r(r+2)} \cdot \frac{1}{r+4}$ மற்றும் $f(r) = \frac{1}{r(r+2)}$ என கொடுக்கப்பட்டுள்ளது.

1. Let's clarify the problem: You are asking why we divide an expression by $4x$ when the expression contains $4y$. 2. Typically, division by $4x$ is done to isolate a variable or

1. **State the problem:** We need to find the value of $\frac{y}{x}$ given the equation $9^x = 81^y$. 2. **Rewrite the bases as powers of the same base:**

1. **State the problem:** You asked, "From where did the 9 come from?" in the expression $\frac{9}{x}$. Let's clarify this. 2. **Recall the original problem:** The original equatio

1. **State the problem:** Given the equation $9^x = 81^y$, find the value of $\frac{9}{x}$. 2. **Rewrite the bases as powers of 3:**

1. **State the problem:** Solve for $x$ in the equation $$16(3)^x = 81(2)^x.$$\n\n2. **Rewrite constants as powers:** Note that $16 = 2^4$ and $81 = 3^4$. Substitute these into the

1. **State the problem:** Solve for $x$ in the equation $16(3)^2=81(2^x)$. 2. **Write down the equation:**

1. **State the problem:** Solve for $x$ in the equation $16(3)^2 = 81(2^{2x})$. 2. **Write down the equation:**

1. The problem is to determine which filter design meets both requirements: - Remove at least 95% of chlorine from tap water.

1. **Problem:** Solve the equation $$6x^4 - 11x^2 + 3 = 0$$ 2. **Formula and approach:** This is a quartic equation but can be treated as a quadratic in terms of $x^2$. Let $y = x^