🧮 algebra

1. The problem is to simplify the fraction $\frac{5}{12}$. 2. The formula for simplifying a fraction is to divide the numerator and denominator by their greatest common divisor (GC

1. **State the problem:** We need to find the number $x$ such that 60 is $\frac{6}{12}$ of $x$. 2. **Write the equation:** Since 60 is $\frac{6}{12}$ of $x$, we write:

1. **State the problem:** We need to analyze and graph the function $$y = |x| - |x - 1|$$. 2. **Recall the definition of absolute value:** For any real number $a$, $$|a| = \begin{c

1. **State the problem:** Find the coordinates of the midpoint of the line segment joining the points $(4, -11)$ and $(-2, 7)$. 2. **Formula for midpoint:** The midpoint $M$ of a l

1. **Problem statement:** Given that one root of the quadratic equation $x^2 - 3x + a = 0$ is $x=6$, find the value of $a$ and the other root. 2. **Formula and rules:** For a quadr

1. **State the problem:** We want to find the order size $x$ that minimizes the cost per unit, given the total cost function $$C(x) = 5x^2 + 320.$$ 2. **Define cost per unit:** The

1. **State the problem:** Simplify the expression $$a^2 (2a + 5a^3 + 1) - 10a$$. 2. **Apply the distributive property:** Multiply $$a^2$$ by each term inside the parentheses:

1. **State the problem:** Multiply the two polynomials $$(x^2 + 2x - 1)(x^2 - x - 4)$$. 2. **Recall the distributive property:** To multiply two polynomials, multiply each term in

1. **State the problem:** Simplify the expression $2 - (3x + 2)$. 2. **Recall the rule:** When subtracting a quantity in parentheses, distribute the minus sign to each term inside.

1. **State the problem:** We need to find how many azzes are in 1 enit given the conversion rates between different coins. 2. **Given conversions:**

1. The problem asks us to compare $12.5 \times 10^{-4}$ and $1.25 \times 10^{-3}$. 2. Both numbers are in scientific notation, which is expressed as $a \times 10^n$ where $a$ is th

1. **State the problem:** Solve the equation $$2(x - 5) + 8x = 20$$ for $x$. 2. **Apply the Distributive Property:** Multiply 2 by each term inside the parentheses:

1. **State the problem:** Solve the equation $2(x - 5) + 8x = 20$ for $x$. 2. **Apply the Distributive Property:** Multiply $2$ by each term inside the parentheses.

1. **State the problem:** Simplify the expression $$\frac{a^{2/3}}{b^{1/3}} + \frac{b^{2/3}}{a^{1/3}}$$ and find which given option it is equivalent to. 2. **Rewrite the expression

1. **Stating the problem:** Simplify the expression $x(12y^2 + 16)x(12y^2 + 16)x(12y^2 + 16)$. 2. **Rewrite the expression:** Notice that the expression is $x(12y^2 + 16)$ multipli

1. **State the problem:** Solve and graph the system of linear equations: $$\begin{cases} X + Y = 2 \\ X - Y = 4 \end{cases}$$

1. The problem is to evaluate the expression $2000(1.10)^{0.5}$. 2. The expression involves multiplying 2000 by the square root of 1.10, since raising to the power 0.5 is equivalen

1. **Stating the problem:** You asked to fill in the JSON response with all the information you gave, which is a request to demonstrate the format and content structure for a math

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

1. The problem asks to express two equations in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 2. The general form $y = mx + b$ means the equation is sol

1. **State the problem:** We need to choose the correct system of equations where $x$ is the number of Christmas ornaments and $y$ is the number of doll dresses, then solve the sys