🧮 algebra

1. The problem is to evaluate the expression $5:2:4 \div 5:4:6$. Here, the colon ":" is interpreted as division, so the expression can be rewritten using division symbols. 2. Rewri

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. Start by isolating the variable term $2x$ on one side. Subtract 3 from both sides:

1. **State the problem:** A man bought a shirt for 50 and sold it for 65. We need to find the percentage profit. 2. **Identify the cost price (CP) and selling price (SP):**

1. The problem asks if $x^{-5}$ can be written as $\frac{1}{x^5}$.\n\n2. Recall the rule of exponents: for any nonzero $x$ and integer $n$, $x^{-n} = \frac{1}{x^n}$.\n\n3. Applying

1. The problem is to understand and analyze the linear equation $x + 2y = 4$. 2. This equation represents a straight line in the coordinate plane.

1. Problem: Express 6 : 25 as a percent. To convert a ratio to a percent, divide the first number by the second and multiply by 100.

1. The question asks to find the values of $x$ and $y$ but does not specify an equation or context. 2. To solve for $x$ and $y$, we need an equation or system of equations involvin

1. **State the problem:** Given $a = \frac{2 - \sqrt{3}}{2 + \sqrt{3}}$ and $b = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}$, find the value of $$a^2 - 5ab + b^2$$ and also compute $a+b$ an

1. The user asked to simplify answers for all questions, but no specific questions were provided. 2. To assist effectively, please provide the algebraic expressions or problems you

1. **State the problem:** Arthur worked for 65 minutes writing 16 words per minute, and Bryony worked for 90 minutes writing 14 words per minute. We need to find who wrote more wor

1. The problem states that the roots of the equation $f(x) = 0$ are given by $$x = \frac{-5 \pm \sqrt{3 - 12k^2}}{4}.$$ We need to find the values of $k$ for which the roots are eq

1. The problem is to express the equation $3x - y = 6$ in terms of $y$. 2. Start by isolating $y$ on one side of the equation.

1. **State the problem:** Rationalise the denominator of $$\frac{6 + 5\sqrt{3}}{3 - 2\sqrt{3}}$$ and express the answer in the form $$a + b\sqrt{3}$$ where $$a$$ and $$b$$ are inte

1. The problem is to analyze the linear function $y=2x+1$. 2. This function is in slope-intercept form $y=mx+b$, where $m=2$ is the slope and $b=1$ is the y-intercept.

1. The problem asks to find all rectangles with natural number dimensions whose area is 72 square units, and then find their perimeters. We also need to determine when the perimete

1. The problem is to analyze the linear function $y=2x+1$. 2. This function is in slope-intercept form $y=mx+b$, where $m=2$ is the slope and $b=1$ is the y-intercept.

1. **State the problem:** We need to show two binomial coefficient identities: a) $\binom{n}{1} = n$

1. The problem is to find an equation for $x$. 2. To provide an equation for $x$, we need more information or context, such as an expression or relationship involving $x$.

1. The problem is to find the Least Common Multiple (LCM) of given numbers using prime factorization. 2. Prime factorization means breaking down each number into its prime factors.

1. The problem asks to find the constant value of either q1 or q2. 2. Since no specific equation or context is given, we assume q1 and q2 are constants.

1. **State the problem:** Solve the system of equations: $$9(9^x) = 27^{y+2}$$