🧮 algebra

1. **State the problem:** We need to find an equation for the total amount of money $y$ the teacher spends on pencils as a function of the number of students $x$.

1. **State the problem:** We want to understand what $8^{1/2}$ means using the properties of integer exponents and define it consistently. 2. **Use the power of a power rule:** The

1. **State the problem:** We need to find the discount percentage when the original price is $7 and the sale price is $3.57. 2. **Formula for discount percentage:**

1. **State the problem:** We need to find the sales tax percentage when a $5 item costs $5.20 after tax. 2. **Formula:** The total price after tax is given by $$\text{Total Price}

1. **State the problem:** Simplify the expression $2\sqrt{45} + 2\sqrt{90} + 3\sqrt{45}$. 2. **Recall the rule:** $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and simplify square

1. **State the problem:** Simplify the expression $$\sqrt{500} + \sqrt{20} + 11\sqrt{5}$$. 2. **Recall the rule:** The square root of a product can be written as the product of squ

1. The problem is to solve the equation $20 = 8$ for $x$. However, this equation as stated is not valid since $20$ does not equal $8$. It seems you want to solve one of the listed

1. **State the problem:** Solve the system of linear equations: $$14p + 4s = 128$$

1. **State the problem:** We are given the system of linear equations:

1. **State the problem:** We are given the sum of the first $n$ terms of a sequence as $$S_n = n(n+1)(n+2)$$ and need to find the 10th term, $a_{10}$. 2. **Recall the formula for t

1. **State the problem:** Graph the quadratic function $y = 6x^2 + 5x - 4$ and find the x-intercepts, y-intercept, axis of symmetry, and vertex. 2. **Formula and rules:**

1. **State the problem:** We are given the height function of an object in terms of time $t$: $$h = -16t^2 + 60t + 3$$ and we want to understand its behavior. 2. **Identify the typ

1. **State the problem:** We need to find what percentage of the total prison population are men. 2. **Identify the given data:**

1. **State the problem:** Given the table of values for $x$ and $y$, find the relationship or function that connects $x$ and $y$. 2. **Observe the data:** The table shows:

1. **State the problem:** We are given a table of values for $x$ and $y$ and need to find an equation that describes the relation between $x$ and $y$.

1. **State the problem:** We want to understand why the variable $x$, representing the side length of squares cut from each corner of a 30 cm by 20 cm cardboard, must be less than

1. **State the problem:** Solve the equation $$4(x+10) = 10x$$ for $$x$$. 2. **Apply the distributive property:** Multiply 4 by each term inside the parentheses.

1. **State the problem:** Solve for $x$ in the equation $$\frac{x}{4} - \frac{x}{5} = \frac{7}{20}.$$\n\n2. **Use a common denominator:** The denominators are 4, 5, and 20. The lea

1. **Problem statement:** Find the 50th term of the sequence 5, 8, 13, 20, ... 2. **Identify the pattern:** The sequence is not arithmetic (difference between terms is not constant

1. The problem: Understand the discriminant rules for quadratic equations. 2. The quadratic equation is generally written as $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are const

1. **Problem (a): Express $16x^2 - 24x + 10$ in the form $(4x + a)^2 + b$.** 2. Start with the expression: