🧮 algebra

1. **Problem statement:** Two hoses together fill a pool in 2 hours. Hose A alone fills the pool in 3 hours. We need to find how long hose B alone takes to fill the pool. 2. **Form

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 6y = -9 \\ -x + 3y = 9 \end{cases}$$

1. **State the problem:** Solve the quadratic equation $y = 2x^2 + 7x - 15$ for $x$ when $y=0$. 2. **Formula used:** To find the roots of a quadratic equation $ax^2 + bx + c = 0$,

1. **Problem 2a:** Express $2x^2 - 8x + 5$ in the form $a(x + b)^2 + c$, where $a$, $b$, and $c$ are integers. 2. The formula to complete the square for a quadratic $ax^2 + bx + c$

1. **State the problem:** We need to determine which expressions equal 80 when $w=4$. 2. **Evaluate each expression:**

1. The problem states that the total cost $y$ including the protection plan is given by the equation $$y = 1.2x$$ where $x$ is the cost of the computer alone. 2. Nathan pays a tota

1. **Solve the system:** $$\begin{cases} 10x - y = 9 \\ -10x + y = -9 \end{cases}$$

1. **State the problem:** Solve the equation $5(u + 4) - 7u = 26$ for $u$. 2. **Apply the distributive property:** Multiply 5 by each term inside the parentheses.

1. **State the problem:** Solve the equation $$4(x - 2) + 2x = -32$$ for $x$. 2. **Apply the distributive property:** Multiply 4 by each term inside the parentheses.

1. The problem is to identify the property used in the equation $A + (-A) = 0$. 2. This equation shows that when you add a number and its additive inverse (the negative of the numb

1. **State the problem:** Solve the system of equations by elimination: $$-v + w = 7$$

1. The problem asks for the focus of the previous problem, which typically involves a conic section such as a parabola. 2. The focus of a parabola given by the equation $y = ax^2 +

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. The formula used here is to isolate $x$ by performing inverse operations.

1. The problem is to solve the equation given by the user. However, no specific equation was provided, so I will demonstrate solving a simple algebraic equation as an example: $2x

1. **State the problem:** Convert the quadratic equation $y = x^2 + 10x + 22$ into vertex form. 2. **Recall the vertex form:** The vertex form of a quadratic is $$y = a(x-h)^2 + k$

1. Problema 40: Considera la matriz $$A = \begin{pmatrix} 1 & \frac{1}{8} & \frac{1}{8} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

1. Planteamos el problema: Calcular $A^{10}$ para la matriz $$A=\begin{pmatrix}0 & a & -b \\ 0 & 0 & b \\ 0 & 0 & 0\end{pmatrix}$$ donde $a,b$ son constantes. 2. Observamos que $A$

1. The problem asks to find the value of the expression $-6k + 5$ when $k = -3$. 2. The formula given is $-6k + 5$. We substitute $k$ with $-3$.

1. The problem asks to find which card matches Card C's simplified expression $x^{-15}$. 2. Card C is $\frac{(x^6)^{-2}}{x^3}$.

1. **State the problem:** Find the greatest common factor (GCF) of the numbers 48 and 136. 2. **Prime factorization:**

1. **State the problem:** Solve the equation $93 + 2k + 96 = 6(-7k + 3) + 35$ for $k$. 2. **Combine like terms on the left side:**