🧮 algebra

1. Stating the problem: We are given the system of equations $$5x + y = 13$$ and $$5xy = 13$$, and asked to find the value of $$12.5x^2 + y^2$$ given that $$xy = 10$$. 2. Step 1: A

1. The problem is to analyze the given equations and solve for the variables involved.\n2. Identify each equation and write down what is known and what needs to be found.\n3. Use a

1. The problem involves finding the total marks for 5 tests given that the mean score for 2 tests is 75. 2. Mean score is calculated as total marks divided by the number of tests.

1. The problem asks to find the fraction of the cake Wetsho left for himself after sharing with his three friends. 2. Wetsho gave Toto \(\frac{1}{6}\), Phatsimo \(\frac{2}{5}\), an

1. The problem asks for the Least Common Multiple (LCM) of the numbers 5, 3, and 6. 2. Recall that the LCM of a set of integers is the smallest positive integer that is divisible b

1. Problem: An object is thrown upward with height given by $$h = -16t^2 + 96t$$. Find $t$ when $h=144$ feet. 2. Set the height equal to 144: $$-16t^2 + 96t = 144$$.

1. Problem (d): Calculate $\frac{2}{3} \times \left(1 \frac{1}{4} - \frac{5}{8}\right)$.\n 2. Convert mixed number to improper fraction: $1 \frac{1}{4} = \frac{5}{4}$.\n

1. **First problem statement:** Find the new quadratic equation whose roots are $\alpha - 2\beta$ and $\beta - 2\alpha$, given the roots $\alpha$ and $\beta$ of $x^2 + 3x - 7 = 0$.

1. Problem a: Calculate the total future profits from a tool whose net return decreases by 10% annually, starting from 400 this year. 2. Since the return decreases by 10% per year,

1. **State the problem:** Solve the equation $$\log_{10}(x - 3) + \log_{10} x = \log_{10} 12$$ for $x$. 2. **Apply logarithm properties:** Recall that the sum of logarithms with th

1. State the problem: Solve the equation $$\frac{3}{2}x + 3 - \frac{1}{2}x + 1 = \frac{1}{x} + 1$$ for $$x$$. 2. Simplify the left side by combining like terms:

1. Stated problem: Solve the equation $$3 \log_{10}{(x-3)} + \log_{10}{x} = \log_{10}{12}$$. 2. Use logarithm property: $$a\log_b{c} = \log_b{c^a}$$, so rewrite the equation as $$\

1. نبدأ بتعريف الدالة $g(x)$، حيث نعلم أن $g(-2)=a$ لبعض القيمة $a$. 2. المطلوب هو حساب قيمة الدالة المركبة $(g \, o \, g \,...\, o g)(-2)$ حيث تم تطبيق $g$ على نفسها $n$ مرات متتا

1. **State the problem:** Find the value of $x$ from the equation $\log_2 x + \log_2 (x-3)$.\n\n2. **Use logarithm properties:** Recall that $\log_a m + \log_a n = \log_a (m \times

1. **Énoncé du problème :** Montrer que

1. Given the equation $x \log_2 x + \log_2 (x - 3) = 2$, we need to find the value of $x$. 2. Recall that $\log_2 (x - 3)$ is defined only if $x - 3 > 0$, so $x > 3$.

1. State the problem: Determine which of the given choices is not in the interval $-\tfrac{1}{2} < x < \tfrac{2}{3}$. 2. Interpretation and method: A number is in the interval if i

1. The problem asks for a pair of factors of -90 whose sum is -1. 2. We start by listing pairs of numbers that multiply to -90.

1. The problem is to determine the relationship between the fractions $\frac{-1}{2}$ and $\frac{2}{3}$ in the context of set inclusion, denoted by $\subseteq$. 2. Set inclusion $A

1. The problem is to find the best approximation interval for $\sqrt{91}$.\n\n2. We know that $9^2 = 81$ and $10^2 = 100$. Since $91$ is between $81$ and $100$, its square root mus

1. Stating the problem: Simplify the expression $$3(6 - x) / (x^3 + 27) + (x+3) / (x+4) - (x-3) / (x^2 - 3x + 9) - (x-4) / (x-3) - 1 / (x+3) + 7 / (x^2 + 3x - 4)$$