🧮 algebra

1. **State the problem:** Solve the equation $12 - 4x = -7x$ for $x$. 2. **Recall the rule:** To solve linear equations, isolate the variable on one side by performing inverse oper

1. The vertex of a parabola given by the quadratic function $y = ax^2 + bx + c$ is found using the formula for the $x$-coordinate of the vertex: $$x = -\frac{b}{2a}$$ 2. Once you f

1. **Problem Statement:** A farmer wants to fence a rectangular area with one side along a stream, so only three sides need fencing. The total fence length is 80 m. We need to find

1. **Problem Statement:** Find the x-intercept and y-intercept of the linear function shown in the graph. 2. **Understanding intercepts:**

1. The problem involves understanding the graph of the function shown, which is a line passing through points approximately (-4,-4) and (4,4). 2. The formula for a line is given by

1. **State the problem:** A tank is initially $\frac{5}{9}$ full of water. After pouring in $6 \frac{3}{4}$ jugs of water, the tank becomes $\frac{2}{3}$ full. We need to find the

1. Let's clarify where the term $8x$ comes from in an algebraic expression. 2. Typically, $8x$ represents a term where 8 is the coefficient and $x$ is the variable.

1. **State the problem:** Solve the equation $19x - 5 - 11x = 15 + 3x$ for $x$. 2. **Combine like terms on the left side:**

1. **State the problem:** Solve for $y$ in the equation $$125y^3 = 8$$. 2. **Formula and rules:** To solve for $y$, isolate $y^3$ by dividing both sides by 125, then take the cube

1. The problem is to simplify the expression $\frac{1}{2}\ln((x-4)^2) + 16$. 2. Recall the logarithm power rule: $\ln(a^b) = b\ln(a)$.

1. The problem asks to identify expressions that represent the verbal phrase: "The difference of 12 and 20% of a number x". 2. The phrase "difference of 12 and 20% of a number x" m

1. The problem asks for the expression representing the difference of 12 and 20% of a number $x$. 2. "Difference of 12 and 20% of $x$" means subtracting 20% of $x$ from 12.

1. **State the problem:** Factor the expression $$\frac{8}{3} - \frac{2}{3}x$$ using the greatest common factor (GCF). 2. **Identify the GCF:** Both terms $$\frac{8}{3}$$ and $$\fr

1. **State the problem:** Subtract the expression $(3.5x - 10)$ from $(4.5x - 5)$, i.e., compute $(4.5x - 5) - (3.5x - 10)$. 2. **Recall the subtraction rule:** When subtracting ex

1. **State the problem:** We need to find which expressions are equivalent to $$- \frac{2}{5}(15 - 20d)$$. 2. **Use the distributive property:** Multiply $$- \frac{2}{5}$$ by each

1. **Solving quadratic equations by completing the square:** The method involves rewriting the quadratic equation in the form $$a(x-h)^2 + k = 0$$ where $h$ and $k$ are constants.

1. **State the problem:** Simplify and evaluate the expression $$\left(\frac{\sqrt{2}+\sqrt{3}+\sqrt{5}}{\sqrt{3}+\sqrt{2}-\sqrt{5}} + \frac{\sqrt{3}+\sqrt{2}+\sqrt{5}}{\sqrt{3}+\s

1. **Stating the problem:** We are given several expressions and equations involving arithmetic operations, fractions, and a function with a square root in the denominator. We need

1. **State the problem:** We need to find Camilla's unit rate of speed, which is the distance she drives per hour, using the graph that shows time in hours and distance in miles. 2

1. **State the problem:** We need to find the rate at which Sebastian and Gabriel complete math problems per minute and then compare these rates to determine who works faster. 2. *

1. We are asked to perform synthetic division on the given polynomials by the specified binomials. 2. Synthetic division is a shortcut method for dividing a polynomial by a binomia