🧮 algebra

1. **State the problem:** We are given data points showing miles driven ($x$) and gallons of gasoline remaining in the tank ($y$). We need to find the slope and y-intercept of the

1. **State the problem:** Solve the cubic equation $$x^3 - 6x^2 + 11x - 6 = 0$$. 2. **Recall the formula and approach:** For cubic equations, one common method is to try factoring

1. **Problem Statement:** We are given the function $$V(t) = 2000 \left(\frac{1}{2}\right)^{\frac{t}{h}}$$ which models exponential decay with an initial value of 2000 and a half-l

1. **State the problem:** Solve the quadratic equation $$5x^2 - 3x + 2 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions a

1. **Problem:** Given the circle equation $$(x+1)^2 + (y-4)^2 = 9,$$ identify the center and radius, and find its x-intercepts. 2. **Formula and rules:** The general form of a circ

1. **Problem Statement:** A man buys a cow for 3000 and sells it for 3600 on credit for 2 years with a simple interest rate of 10% per annum. We need to find if he made a gain or l

1. The problem asks for an interpretation of each solution to a given equation or system. 2. Typically, solutions to equations represent values that satisfy the equation, meaning w

1. **State the problem:** Simplify the expression $90x + x$. 2. **Formula and rules:** When adding like terms, add their coefficients and keep the variable the same.

1. The problem is to simplify an expression or equation (not specified by the user). 2. To simplify an expression, we combine like terms, apply arithmetic operations, and use algeb

1. We are asked to simplify the expression $\left(100000 x^5 y^8\right)^{\frac{1}{3}}$. 2. The formula to simplify an expression raised to a fractional power is $\left(a^m\right)^n

1. The problem is to evaluate the expression $30 \times 3 \times t$ where $t$ is a variable. 2. The formula used here is the multiplication of constants and a variable: $a \times b

1. **State the problem:** Simplify the expression $$\frac{(x^{-2} \cdot x^{4})^{3}}{(x^{5} \cdot x^{-3})^{-2}}$$. 2. **Recall the exponent rules:**

1. **State the problem:** Solve the cubic equation $$6x^3 + 25x^2 - 24x + 5 = 0$$ given that one rational root is 5. 2. **Use the Factor Theorem:** Since 5 is a root, $(x - 5)$ is

1. **Problem:** Show that if the linear equations $x_1 + kx_2 = c$ and $x_1 + lx_2 = d$ have the same solution set, then the two equations are identical (i.e., $k = l$ and $c = d$)

1. **Problem Statement:** Given two functions $f(x) = 2x$ and $g(x) = x + 9$, find: a. $(f \circ g)(x)$

1. **State the problem:** Solve the system of equations: $$\begin{cases} 2x + 3y - z = 5 \\ x - y + 4z = 2 \\ 3x + 2y + z = 7 \end{cases}$$

1. The problem is to factor the expression $x^2 - 16$. 2. Recognize that this is a difference of squares, which follows the formula $a^2 - b^2 = (a - b)(a + b)$.

1. The question asks about the hardest equation to factor. 2. Factoring difficulty depends on the type and degree of the polynomial.

1. The problem: Understand the basic rules for factoring algebraic expressions. 2. Factoring is the process of breaking down an expression into simpler expressions (factors) that,

1. The problem asks which graph represents the solution set of the inequality $y > 4x - 3$. 2. The boundary line is given by the equation $y = 4x - 3$.

1. **State the problem:** We need to find the graph that best represents the solution set for the system of inequalities: $$y \leq -\frac{2}{3}x - 2$$