🧮 algebra

1. **State the problem:** Solve the system of linear equations using the inverse matrix method: $$\begin{cases} 2x - 3y = -1 \\ -3x + 5y = 3 \end{cases}$$

1. **State the problem:** Simplify the expression $$(2x+3)(x+1)+(3+2x)(x-2)$$. 2. **Use the distributive property (FOIL) to expand each product:**

1. المشكلة: تبسيط التعبير الجبري المعتدل. 2. القاعدة: لتبسيط التعبير المعتدل، نجمع الحدود المتشابهة ونرتبها حسب القوى.

1. **State the problem:** Evaluate the expression $16 \div 4 + 8 \times 3$ correctly. 2. **Recall the order of operations (PEMDAS/BODMAS):**

1. **State the problem:** Evaluate the expression $16 \div 4 + 8 \times 3$. 2. **Recall the order of operations:** According to PEMDAS/BODMAS, division and multiplication are perfo

1. The problem is to graph the function $y=2x^{3}$. 2. The function is a cubic polynomial where the coefficient 2 scales the cubic term $x^{3}$.

1. **State the problem:** We want to express the function $$f(x) = \frac{9x^2 + 4}{(2x + 1)(x - 2)^2}$$ as a sum of partial fractions. 2. **Recall the formula and rules:** For a ra

1. The problem is to solve the equation $$y(600 - t)^{-5} = 3(600 - t)^{-4} + C$$ for $y$. 2. To isolate $y$, multiply both sides of the equation by $(600 - t)^5$ to eliminate the

1. **Problem:** Solve the equation $$3x^2 + 2x - \sqrt{3x^2 + 2x - 1} = 3$$. 2. **Step 1: Isolate the square root term.**

1. **Stating the problem:** We want to verify the inequality and the recurrence relation given: $$U_{n+1} < \frac{2}{5} U_n \quad \forall n$$ 2. **Given expressions:** The user man

1. Let's clarify why we set $a-2=0$ in some problems. 2. Usually, this happens when we want to find the value of $a$ that makes an expression equal to zero or satisfies a certain c

1. **State the problem:** Given the quadratic function $f(x) = (a-2)x^2 + (b+3)x + (c-2)$, we are told it is zero degree and its range is 5. We need to find $a+b+c$. 2. **Understan

1. **State the problem:** Find the value of $k$ such that $\frac{p}{x+r} + \frac{q}{x-r} = \frac{k}{2x}$.\n\n2. **Write the equation:** $$\frac{p}{x+r} + \frac{q}{x-r} = \frac{k}{2

1. **State the problem:** Solve the equation $$2^{-2x+5} \times \frac{1}{4} = 2^{x+2}$$ for $x$. 2. **Rewrite the equation using properties of exponents:** Note that $\frac{1}{4} =

1. **State the problem:** We are given two equations: $$P_A = \frac{30 + P_B}{3}$$

1. The problem is to simplify the expression $5p + 3 - 2p + 8$. 2. We use the rule that like terms (terms with the same variable) can be combined by adding or subtracting their coe

1. The problem is to understand and calculate combinations, often denoted as $nCr$, which represent the number of ways to choose $r$ elements from a set of $n$ elements without reg

1. The problem involves the ratio $5:7$ and some numbers $120$, $4$, and $3$, with variables $x$ and $y$. We need to find the values of $x$ and $y$ that satisfy the given ratio and

1. **Stating the problem:** We need to find the value of the expression $$\frac{6 + \sqrt{5}}{6 - \sqrt{5}} \times \frac{6 - \sqrt{5}}{6 + \sqrt{5}}$$.

1. The problem is to analyze the function $y = x^2 + 1$. 2. This is a quadratic function in the form $y = ax^2 + bx + c$ where $a=1$, $b=0$, and $c=1$.

1. **State the problem:** Find the slope of the line passing through the points $(-1,4)$ and $(4,-3)$. 2. **Formula for slope:** The slope $m$ of a line through points $(x_1,y_1)$