🧮 algebra

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

1. The problem is to add the fractions $\frac{2}{3}$ and $\frac{5}{7}$. 2. To add fractions, we need a common denominator. The denominators here are 3 and 7.

1. **Problem statement:** Calculate the revenue from only drinks given the total revenue and percentages of revenue from entries, drinks, and food, including their intersections.

1. **State the problem:** We have a parabola defined by the height function $$h = -0.5d(d - 24)$$ where $h$ is the height in meters and $d$ is the horizontal distance in meters. 2.

1. **State the problem:** We are given the height of the water stream as a function of horizontal distance $d$ from the hose: $$h = -0.5 d (d - 24)$$ We need to find the height whe

1. **State the problem:** We are given the height of a water stream from a fire hose as a function of horizontal distance $d$: $$h = -0.5d(d - 24).$$ We need to find the horizontal

1. **State the problem:** We are given the function $h = -0.5d(d-24)$ which describes a parabola. We want to understand its shape, vertex, and intercepts. 2. **Rewrite the function

1. **State the problem:** Solve for $x$ in the equation $$(5x + 6)(3x - 8) = 0.$$ 2. **Formula and rule:** If a product of two factors equals zero, then at least one of the factors

1. **State the problem:** Solve for $x$ in the equation $$(x - 3)(x - 8) = 0.$$\n\n2. **Formula and rule:** If a product of two factors equals zero, then at least one of the factor

1. **State the problem:** Solve for the two possible values of $x$ in the equation $$(x - 3)(x - 8) = 0.$$\n\n2. **Formula and rule:** The zero product property states that if the

1. **State the problem:** Evaluate $\log_{11}{121}$. 2. **Recall the logarithm definition:** $\log_a{b} = c$ means $a^c = b$.

1. **State the problem:** Evaluate $\log 10^{1000}$. 2. **Recall the logarithm power rule:** $\log a^b = b \log a$. This means we can bring the exponent down as a multiplier.

1. **State the problem:** Calculate the sum of the integers from $i=5$ to $i=7$, i.e., compute $\sum_{i=5}^7 i$. 2. **Formula used:** The summation of consecutive integers from $a$

1. The problem is to find the value of $\log_2(20)$.\n\n2. Recall the definition of logarithm: $\log_b(a)$ is the exponent to which the base $b$ must be raised to get $a$.\n\n3. We

1. **Stating the problem:** We are given the equation $$\frac{2x+3}{x-1} = 4$$ and need to solve for $x$. 2. **Formula and rules:** To solve rational equations, multiply both sides

1. **State the problem:** Factor the quadratic expression $$147m^2 - 168m + 48$$ completely. 2. **Identify the greatest common factor (GCF):** The coefficients are 147, -168, and 4

1. **State the problem:** Factor the quadratic expression $$81v^2 + 198v + 121$$ completely. 2. **Recall the factoring formula:** For a quadratic $$ax^2 + bx + c$$, we look for two

1. **State the problem:** Solve the system of linear equations: $$-x + 3y = 5$$

1. The problem is to find the vertex of the quadratic function $$y = -3x^2 + 12x + 29$$. 2. The vertex of a quadratic function in the form $$y = ax^2 + bx + c$$ can be found using

1. **State the problem:** Simplify the expression $$(4x^3 + 3x - 1) - (-7x^3 + 2x^2 + 3x - 1 + 7x^3 - 2x^2 + 10)$$ 2. **Rewrite the expression inside the parentheses:**

1. **State the problem:** Solve the system of linear equations: $$20 = 2x - y$$