🧮 algebra

1. **Problem:** Identify if the given statements about the rational number system are True or False, and correct the false ones. 2. **Formula/Rules:**

1. **Planteamiento del problema:** Encontrar la ecuación de la recta que pasa por el punto $(1,7)$ y tiene pendiente $\frac{2}{3}$. 2. **Fórmula a usar:** La ecuación punto-pendien

1. **State the problem:** Divide and reduce to lowest terms the mixed numbers: $$1 \frac{1}{3} \div 2 \frac{5}{6}$$ 2. **Convert mixed numbers to improper fractions:**

1. **State the problem:** Multiply $1000000000000000000000$ by $10000000000000000$. 2. **Use the multiplication rule for powers of 10:** Each number is a power of 10. We can write

1. **State the problem:** Solve the equation $$\frac{2x - 3}{x^2 - 4} + \frac{x + 1}{x^2 - x - 2} = \frac{4}{x - 2}$$ for $x$. 2. **Factor denominators:**

1. **State the problem:** Simplify the expression $$\frac{3x}{5} + \frac{4x+7}{5}$$. 2. **Formula and rules:** When adding fractions with the same denominator, add the numerators a

1. **State the problem:** We are given the equation of a circle: $$x^2 + y^2 - 14x + 10y + 49 = 0$$

1. **State the problem:** We need to find Karen's mother's present age given the equation formed by the problem statement. 2. **Translate the problem into an equation:** Let $x$ be

1. **State the problem:** Add the fractions $\frac{1}{2}$, $\frac{2}{5}$, and $\frac{9}{10}$. 2. **Formula and rules:** To add fractions, they must have a common denominator. Find

1. The problem is to understand the meaning and usage of the variable $d$ in a mathematical context. 2. In algebra and geometry, $d$ often represents a distance, difference, or a v

1. **Stating the problem:** The distance going is 5 km with a speed of 40 km/h. The return time is 10 minutes less than the going time.

1. **State the problem:** Calculate the value of $9^{900}$. 2. **Recall the formula:** For any number $a$ and exponent $n$, $a^n$ means multiplying $a$ by itself $n$ times.

1. The problem is to solve the equation or expression given by the user. However, no specific equation or expression was provided. 2. Since no equation is given, we cannot apply an

1. **State the problem:** Solve for all real values of $x$ in the equation $$8^x + 2^x = 130.$$\n\n2. **Rewrite the bases:** Note that $8 = 2^3$, so we can write $$8^x = (2^3)^x =

1. **State the problem:** We are given the polynomial $$40\sqrt{5} x^{3} - 3\sqrt{3} y^{3}$$ and it is factored as $$(2\sqrt{5} x - \sqrt{3} y)(Ax^{2} + Bxy + Cy^{2}).$$ We need to

1. **State the problem:** We are given the polynomial $$405\sqrt{x^3} - 33\sqrt{y^3}$$ and it is factored as $$(25\sqrt{x} - 3\sqrt{y})(Ax^2 + Bxy + Cy^2).$$

1. **State the problem:** Solve the linear equation $3x + 6 = 2x - 7$ for $x$. 2. **Formula and rules:** To solve for $x$, we use the properties of equality: we can add, subtract,

1. **State the problem:** Solve the linear equation $3x + y = 33$ for $y$ in terms of $x$. 2. **Formula and rules:** To isolate $y$, subtract $3x$ from both sides of the equation.

1. The problem is to find the range of numbers from 0 to 100. 2. This is a simple interval notation problem where the set includes all numbers starting at 0 and ending at 100.

1. **Planteamiento del problema:** Tenemos una parábola con vértice en $(-1,-2)$ y que pasa por varios puntos dados. Queremos encontrar la ecuación de la forma $$y = ax^2 + bx + c$

1. **State the problem:** We need to find another way to express the expression $$12(t - 0.25)$$ where $t$ is the time in hours. 2. **Use the distributive property:** The distribut