🧮 algebra

1. **State the problem:** Find the equation of the line perpendicular to $y = \frac{1}{2}x + 8$ that passes through the point $(5, 3)$. 2. **Recall the formula:** The point-slope f

1. The problem asks for the equation of a line perpendicular to $y=\frac{1}{6}x+7$ that passes through the point $(2,6)$, expressed in point-slope form. 2. Recall the point-slope f

1. **Problem statement:** Calculate the values of $x = \sqrt{9 - 4\sqrt{5}}$ and $y = \sqrt{9 + 4\sqrt{5}}$, then evaluate and compare the expression $(9 + \sqrt{5})^2 - (9 - \sqrt

1. **Problem Statement:** Identify the key features and equation of the exponential function passing through points (0,4) and (1,20) with horizontal asymptote $y=0$. 2. **General F

1. **Problem statement:** Calculate the value of the piecewise function $$Z = \begin{cases} \sqrt{x^2 + 2x - 1 - 3y + 2y^3}, & x > y \\ |2x^3 + 4y^2| - 3x^4, & x < y \\ \sqrt[3]{4y

1. **Problem statement:** Calculate the value of the piecewise function $$Z = \begin{cases} 8x^3 + 4xy - xy^2, & x < y \\ |3x^4 - 5y^3| - 12xy, & x = y \\ \sqrt[3]{3} \sin(4x) \cdo

1. **Problem 50(a):** Find an equation connecting total profit $P$ and selling price $x$ given that $P$ partly varies directly as $x$ and partly as $x^2$. 2. Since $P$ varies partl

1. **State the problem:** We need to find the cube root of the expression $$\sqrt[3]{\frac{3.416 \times 0.0789}{\frac{691.6 \times 1.41}{1000}}}$$ using logarithm tables. 2. **Rewr

1. **State the problem:** We need to evaluate the expression $$\sqrt{\frac{3.416 \times 0.0789}{\frac{691.6 \times 1.41}{1000}}}$$ using logarithm tables. 2. **Recall the logarithm

1. The problem is to solve an equation or expression using logarithm tables. 2. Logarithm tables help us find the logarithm (log) of numbers, which can simplify multiplication, div

1. **State the problem:** We need to evaluate the expression $$\sqrt{\frac{3.416 \times 0.0789}{\frac{691.6 \times 1.41}{1000}}}$$ using mathematical tables. 2. **Rewrite the expre

1. **State the problem:** Solve the linear equation $3y + 5 = 2y + 20$ for $y$. 2. **Write down the equation:**

1. **State the problem:** We have two separate equations involving fractions:

1. Let's clarify the concept of "exact radius." When a problem asks for the exact radius, it means you should provide the precise value, including any decimals or radicals, without

1. The problem is to solve or analyze "test 2" visually, but since no specific equation or function is given, I will explain how to approach a visual presentation for a typical alg

1. **Problem:** Use difference tables to find the next term in each sequence. **Sequence 1:** 2, 5, 8, 11, 14, ...

1. The user request is vague: "do the maths". Since no specific problem is given, I will demonstrate a simple algebraic problem and solution. 2. Problem: Solve the quadratic equati

1. **Problem Statement:** Use synthetic division to find the zeros and factors of the polynomial equation. 2. **General Approach:** Synthetic division is a shortcut method for divi

1. **State the problem:** Simplify the expression $-(4x^3 + 3x^2 - 3x + 12)$. 2. **Formula and rule:** When a negative sign is placed before parentheses, it means to distribute the

1. **Problem a:** Solve the equation $$\sqrt{4x - 7} + 2 = 5$$ 2. **Step 1:** Isolate the square root term:

1. **Problem statement:** Pipe A fills a tank in 3 hours, pipe B fills the same tank in 6 hours, and pipe C empties the tank in 8 hours. Pipes A and B start filling the empty tank