🧮 algebra

1. **State the problem:** We are given a piecewise function:

1. **State the problem:** Evaluate the expression $84(12-5)\cdot 2 + 4$. 2. **Apply the order of operations (PEMDAS/BODMAS):**

1. **State the problem:** We need to find the cost per tree for two types of Christmas trees: Noble Fir (n) and Douglas Fir (d). 2. **Write the system of equations from the problem

1. **State the problem:** A car and a motorcycle start from the same place at noon, traveling in the same direction. 2. **Define variables:** Let $m$ be the speed of the motorcycle

1. **State the problem:** We need to find the value of $x$ in the equation $$68 + 3x + 25 + x = 141$$ which represents the total number of students. 2. **Combine like terms:** Add

1. **State the problem:** Multiply out and simplify the expression $ (x + 3)(x - 2) $. 2. **Formula used:** To multiply two binomials, use the distributive property (also known as

1. **State the problem:** Solve the quadratic equation $$13x^2 + 10x - 3 = 0$$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0

1. Let's create an equation where all terms cancel out at the end. 2. Consider the equation $$\frac{x^2 - 1}{x - 1} = x + 1$$.

1. **State the problem:** We are given that \((2x + 3)\) is a factor of the cubic polynomial \(4x^3 - 12x^2 - 7x + 30\). We need to find all three solutions to the equation \(4x^3

1. **State the problem:** Multiply out and simplify the expression $$(4x - 10\sqrt{x})(2x + 5\sqrt{x} - 7).$$ 2. **Recall the distributive property:** To multiply two expressions,

1. **Problem statement:** (a) Find the highest common factor (HCF) of $5A$ and $2B$ given:

1. **State the problem:** Find the value of $k$ for which the system of equations $$\begin{cases} kx - y = 2 \\ 6x - 2y = 3 \end{cases}$$

1. **State the problem:** Find the equation of the perpendicular bisector of the line segment AB where A = (17,9) and B = (23,39). 2. **Formula and rules:**

1. **State the problem:** Simplify or analyze the quadratic expression $$x^2 - 4dx + 2d^2$$ where $x$ and $d$ are variables. 2. **Recall the quadratic form:** A quadratic expressio

1. Il problema richiede di risolvere l'equazione quadratica $$x^2 - 4x + 2p^2 = 0$$ dove $p$ è una costante. 2. La formula generale per risolvere un'equazione quadratica $$ax^2 + b

1. **State the problem:** Find the value of $k$ for which the system of equations $$kx - y = 2$$

1. **State the problem:** We have a pipe network with nodes E, D, A, B, C and flows $x_1$ through $x_6$ between them. Inputs are 200 at E, 150 at D, 25 at A, and output is 200 at C

1. **Problema 4: Identificar les antiimatges de la funció $f(x) = 3x + 2$ per a diferents valors de $y$.** 2. La funció és $f(x) = 3x + 2$. Per trobar l'antiimatge d'un valor $y$,

1. **Calcula la imatge f(x) per a cada valor de x** **a.** $f(x) = 2x + 1$

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. The formula used here is to isolate the variable $x$ by performing inverse operations.

1. **Stating the problem:** Simplify the expression $$L = \frac{7}{6} \times \left(\frac{7}{2} - \frac{3}{2}\right)$$. 2. **Apply the subtraction inside the parentheses:**