🧮 algebra

1. **State the problem:** We need to find the dimensions of a rectangular field with one side as a gate of length $12m$ and the rest fenced with $1000m$ of fencing material. We wan

1. The problem is to substitute $3\cos 2x$ with a new variable $z$ to simplify expressions or equations. 2. The substitution rule allows us to replace a complex expression with a s

1. **State the problem:** We are given the remainders when $W(x)$ is divided by $(x-500)$ and $(x+400)$, and we want to find the remainder when $W(x)$ is divided by $x^2 - 100x - 2

1. **State the problem:** Solve the equation $2 = 2 + 5$. 2. **Analyze the equation:** The equation states that $2$ is equal to $2 + 5$.

1. **State the problem:** We need to find the equation of line L which crosses the y-axis at $3$ and passes through the point $(1, -1)$. 2. **Recall the slope-intercept form:** The

1. **State the problem:** We need to find the points where the graphs of $$y = x^2 - 5x - 3$$ and $$y = x - 4$$ intersect. 2. **Set the equations equal:** Since both equal $$y$$, s

1. **State the problem:** We need to find how many solutions the equation $$-5 = |4w| - 5$$ has. 2. **Isolate the absolute value expression:** Add 5 to both sides:

1. **State the problem:** Evaluate $$\left(-\frac{2}{3}\right)^3 \cdot \frac{1}{2}$$ and simplify the answer. 2. **Recall the formula:** When raising a fraction to a power, raise b

1. **State the problem:** We need to find how many solutions the equation $$|-6n| + 3 = 15$$ has. 2. **Recall the absolute value property:** The absolute value of any number is alw

1. **State the problem:** We need to evaluate the expression $$\left(-\frac{2}{3}\right)^2$$. 2. **Recall the rule:** When squaring a fraction, square both the numerator and the de

1. **State the problem:** Find the mean of the numbers $$-3, -5, 6, 4, 5, 4, -4$$. 2. **Formula for the mean:**

1. **State the problem:** Find the $y$-intercept of the function $$y = x(x-1)(x+4)$$. 2. **Recall the rule for $y$-intercept:** The $y$-intercept occurs where $x=0$. To find it, su

1. **State the problem:** We are given that a mug is $\frac{2}{7}$ full and contains $\frac{5}{7}$ of a cup of water. We need to find the total capacity of the mug. 2. **Formula us

1. **Problem Statement:** Compute the determinant of a matrix using cofactor expansion across the first row. Four options (A, B, C, D) are given with different expansions. 2. **Rec

1. **State the problem:** We want to compute the determinant of the matrix

1. **State the problem:** We need to find the number of pints of blackberries in a purchase where the total cost is $37.00 at Store A and $66.00 at Store B. 2. **Define variables:*

1. **State the problem:** We need to find the value of $$k = a^2 + \frac{9(b-2)}{4c}$$ given $$a = -2.4$$, $$b = 3.3$$, and $$c = -1.9$$. 2. **Write the formula:** $$k = a^2 + \fra

1. **State the problem:** We need to find the weight of the new wheelchair wheel which is $27\%$ lighter than the old wheel weighing $840$ grams.

1. **State the problem:** We are given two functions:

1. **State the problem:** Simplify the expression $$X2-3$$. 2. **Interpret the expression:** Assuming $$X2$$ means $$X \times 2$$ or $$2X$$, the expression becomes $$2X - 3$$.

1. **State the problem:** We are given the equations $5^p = 9$, $9^q = 12$, $12^r = 16$, $16^s = 20$, and $20^t = 25$. We need to find the value of the product $pqrst$. 2. **Expres