🧮 algebra

1. The problem is to prove a mathematical statement or theorem. Since the user did not specify which proof is needed, I will demonstrate a common algebraic proof: the distributive

1. **State the problem:** We need to find which point on the number line best represents the fraction $\frac{1}{3}$. 2. **Recall the value of $\frac{1}{3}$:** The fraction $\frac{1

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. The formula used here is to isolate $x$ by performing inverse operations. We subtract 3 from both sides and then d

1. **State the problem:** Solve the inequality $|x - 3| \leq 12$ for $x \in \mathbb{R}$. 2. **Recall the definition and rule for absolute value inequalities:** For any real number

1. **State the problem:** Solve the two-step equation $3x + 4 = 19$ and check the solution. 2. **Formula and rules:** To solve two-step equations of the form $px + q = r$, first is

1. **State the problem:** Express the given number as a power of 3. 2. **Formula and rules:** A number expressed as a power of 3 is written as $3^n$ where $n$ is an integer.

1. **State the problem:** Calculate the value of $3^{10} + 3^{10} + 3^{10}$. 2. **Use the properties of exponents and addition:** Since all terms are the same, we can factor out $3

1. **State the problem:** Solve the equation $$\sqrt{\frac{x}{x+3}} - \sqrt{\frac{x+3}{x}} = 2.$$\n\n2. **Identify domain restrictions:** Since we have square roots and denominator

1. **State the problem:** Solve the proportion $1,11 = \frac{1}{x}$ for $x$. 2. **Rewrite the problem:** The comma in $1,11$ likely represents a decimal point, so the problem is $1

1. **Problem statement:** We are given a vector $r = \begin{pmatrix}80 \\ 50\end{pmatrix}$ and a vector $x = \begin{pmatrix}30 \\ 20\end{pmatrix}$. We need to analyze these vectors

1. **Problem:** Löse die Gleichung $$6x - \left(\frac{3}{5} + 2x\right) = x - \frac{3}{4}$$ 2. **Formel und Regeln:** Um Bruchgleichungen zu lösen, multiplizieren wir beide Seiten

1. **State the problem:** We are given points $A(-12,-9,k+3)$ and $B(-15,-9,3k)$, and the midpoint of segment $AB$ lies in the $xy$-plane. We need to find the value of $k$. 2. **Re

1. **State the problem:** Solve the equation \(5^{2n} \times 25^{2n-1} = 625\). 2. **Recall the rules and formulas:**

1. **Problem 9 (i): Solve the equation** $$2^{2x} - 9(2^x) + 8 = 0$$ 2. **Rewrite using substitution:** Let $$y = 2^x$$. Then $$2^{2x} = (2^x)^2 = y^2$$.

1. **State the problem:** Find the equation of the line perpendicular to the line $$y=\frac{1}{3}x$$ and passing through the point $$(2,5)$$. 2. **Recall the rule for perpendicular

1. **State the problem:** Graph the function $$y=\left(\frac{1}{2}\right)^{x-1} - 10$$ using the given table of values and identify the asymptote.

1. **State the problem:** Graph the function $$y=\left(\frac{1}{2}\right)^{x+1} - 7$$ using the given table of values and identify the horizontal asymptote.

1. **State the problem:** Solve the simultaneous equations: $$x + y + z = 16$$

1. **State the problem:** Simplify the expression $$(2x-3)(2x-3)^2$$. 2. **Understand the expression:** The expression is a product of $(2x-3)$ and $(2x-3)^2$. Using the laws of ex

1. **State the problem:** Evaluate the expression $4 + 5^3 - 3^2 + 8(27 \div 9)^2$. 2. **Recall order of operations:** Use PEMDAS (Parentheses, Exponents, Multiplication and Divisi

1. **State the problem:** Evaluate the expression $7^3 - 3(-4)^3$ and verify if the handwritten result $535$ is correct. 2. **Recall the formulas and rules:**