🧮 algebra

1. The user input is incomplete since the value of $z$ is not provided. 2. To proceed with any calculations or explanations, please provide the full expression or value for $z$.

1. The problem states a geometric pattern where a figure's number of sides increases linearly, starting with a triangle (3 sides) and adding 2 sides for each consecutive figure. 2.

1. We are given the equation $a - b = c^2$. 2. This equation states that the difference between $a$ and $b$ equals the square of $c$.

1. The problem involves analyzing a Grade 4 learner's interpretation of the number sentence $$12 + 5 = \square + 8$$ and their incorrect working: $$12 + 5 = 17 + 8 = 25$$. 2. The l

1. We are asked to express the expression $$\frac{y^2}{y^2 - x^2} - \frac{y}{y - x}$$ as a single simplified fraction. 2. Start by recognizing that the denominator $$y^2 - x^2$$ is

1. Let's solve the system by adding the equations to eliminate one variable. 2. Consider the system:

1. Stating the problem: Solve the system of equations $$5x - 4y = 11$$

1. The problem is to solve the quadratic equation $$x^2 - 2x = 0$$. 2. Start by factoring the equation. Factor out the common factor $$x$$:

1. נתון: $f(x)=4x+12$ ו- $g(x)=\frac{1}{f(x)}=\frac{1}{4x+12}$.\n\n(א) תחום ההגדרה של $g(x)$ הוא כל הערכים שבהם המכנה שונה מאפס, כלומר: $$4x+12 \neq 0 \Rightarrow x \neq -3.$$\nלכן

1. **State the problem:** We need to eliminate the denominators in the equation $$\frac{2}{3} + 1 = \frac{3}{2}$$. 2. **Identify denominators:** The denominators are 3 and 2.

1. The problem asks to find equivalent fractions for each given fraction with specific denominators. 2. For each fraction $\frac{a}{b}$, find the factors to get the new denominator

1. The problem asks to find the inverse function of $f(x) = x + \frac{2}{3}$.\n\n2. To find the inverse function $f^{-1}(x)$, start by replacing $f(x)$ with $y$: $$y = x + \frac{2}

1. We are asked to multiply $3.4 \times 10^2$ by $4 \times 10^4$. 2. Multiply the decimal parts: $3.4 \times 4 = 13.6$.

1. The problem is to solve the equation $$3x = x - 9$$ for the variable $$x$$. 2. Start by subtracting $$x$$ from both sides to get all terms with $$x$$ on one side:

1. Calculate $6^5$. This means multiplying 6 by itself 5 times: $$6^5 = 6 \times 6 \times 6 \times 6 \times 6.$$\n2. Compute step-by-step: $$6 \times 6 = 36,$$ $$36 \times 6 = 216,

1. We are given the equation $1252^x = 1$ and asked to solve for $x$. 2. Recall that any nonzero number raised to the power $0$ is equal to $1$, so $a^0 = 1$ for $a \neq 0$.

**Problem 1:** Given matrices $$A=\begin{bmatrix}3 & 2 \\ -1 & 5\end{bmatrix}, \quad B=\begin{bmatrix}2 & -4 \\ 3 & 1\end{bmatrix}$$

1. The problem asks to find the real solutions of the inequality $x - 3 \leq 2$ where $x$ is the real part of the complex number $z = x + ij$. 2. To solve the inequality, isolate $

1. Stating the problem: Solve the quadratic equation $x^2 + 4x + 4 = 0$.\n\n2. Recognize this is a quadratic equation in standard form $ax^2 + bx + c = 0$ where $a=1$, $b=4$, and $

1. The problem is to solve the quadratic equation $x^2 + 4x + 4 = 0$. 2. First, recognize that the quadratic can be factored as a perfect square trinomial.

1. The problem involves plotting multiple quadratic equations to form a butterfly shape. 2. Each equation represents a part of the butterfly: right wing, left wing, upper and lower