🧮 algebra

1. **Problem statement:** A group of 12 people have enough water to last 9 days. (a) If 6 more people join, how long will the water last?

1. Problem 16a: Find values of $m$ such that the quadratic equation $$x^2 + (3 - m)x + 2m - 1 = 0$$ has real roots. 2. For real roots, the discriminant must be non-negative:

1. The problem is to analyze and correct the graph of the function $f(x) = 8 \cdot \left(\frac{5}{4}\right)^x$. 2. This is an exponential function with base $\frac{5}{4} > 1$, so t

1. **State the problem:** Solve the equation $$\frac{1}{x} - 1 = \frac{3e^{3x}}{x+3} - \frac{e^{3x}}{(x+3)^2}$$ for the values $x = -2.75$, $x = -2$, and $x = -1$. 2. **Substitute

1. **State the problem:** We have a quadratic function $f(x) = ax^2 + bx + c$ with vertex at $(0.5, -12.5)$ and x-intercepts at $(2, 0)$ and $(p, 0)$. We need to find $p$, the coef

1. **State the problem:** We have a cylindrical tin can with height function $h(t) = 3t^2 + 0.5$ for $4 \leq t \leq 14$. We want to find the height when $t$ is given. 2. **Calculat

1. The problem is to solve the equation $7 \frac{b+5}{10} = \frac{5}{7}$ for $b$. 2. Start by writing the equation clearly:

1. The problem is to solve the equation $\frac{7}{n} = \frac{8}{7}$ for $n$. 2. To solve for $n$, cross-multiply the fractions:

1. Stating the problem: Solve the equation $\frac{7}{b} + 5 = \frac{10}{5}$ for $b$. 2. Simplify the right side: $\frac{10}{5} = 2$.

1. Problem: Calculate the percentage error in measuring a string of length 400 cm when rounded to the nearest metre and nearest 100 m. Step 1: Convert 400 cm to metres: $$400\text{

1. Сформулюємо умову задачі: вектори $\mathbf{a} = (5, 3, x)$ та $\mathbf{b} = (y, 6, 4)$ перпендикулярні, тобто їх скалярний добуток дорівнює нулю: $$\mathbf{a} \cdot \mathbf{b} =

1. **State the problem:** Simplify the expression $$\frac{1}{2} + \frac{3}{4} - \frac{1}{3} \times \frac{2}{5} \div \frac{4}{7} + \frac{2}{6}$$. 2. **Handle multiplication and divi

1. **State the problem:** Simplify the expression $$\sqrt{\frac{4x}{10}} \div \sqrt{\frac{10}{9x}}$$. 2. **Rewrite the division of square roots as a single square root:**

1. **State the problem:** We are given the system of equations: $$3x^2 + 2xy + 4y + 2y^2 = 4$$

1. **Problem 8:** Alvin and Simon share 540 in ratio 4:5. Alvin's share = $\frac{4}{4+5} \times 540 = \frac{4}{9} \times 540 = 240$. Simon’s share = $\frac{5}{9} \times 540 = 300$.

1. **State the problem:** A company sells a product at 1200 per unit and charges a delivery fee of 200 per product. If the total order value plus delivery fees exceeds 5000, a 4% d

1. Problem 4: Carly and James share money in ratio 5:3. Carly gets 70 more than James. Find James' amount. 2. Let James' amount be $x$. Then Carly's amount is $x + 70$.

1. **Problem:** Molly, Paige, and Demi share 42 sweets in the ratio 3 : 2 : 1. Find how many sweets each receives. 2. **Step 1:** Add the parts of the ratio: $3 + 2 + 1 = 6$ parts

1. **Question 2: Molly, Paige and Demi share 42 sweets in the ratio 3 : 2 : 1.** Step 1: Add the parts of the ratio: $3 + 2 + 1 = 6$ parts total.

1. **Problem 2:** Molly, Paige, and Demi share 42 sweets in the ratio 3 : 2 : 1. 2. First, find the total parts of the ratio: $3 + 2 + 1 = 6$ parts.

1. **State the problem:** Determine whether the equation $x - 4 = -4 + x$ has one solution, no solutions, or infinitely many solutions. 2. **Simplify both sides:** The left side is