🧮 algebra

1. The problem states the equation: $3=3$. 2. This is an identity because both sides of the equation are exactly equal.

1. Problem statement: We are asked to find a function $D(t)$ that describes the temperature difference between a hot cake and the cooler after $t$ minutes. 2. Given: Initial temper

1. **State the problem:** Solve the equation $$x - 7y = 2x + \frac{1}{2}$$ for one variable in terms of the other. 2. **Rewrite the equation:** $$x - 7y = 2x + \frac{1}{2}$$

1. We are given the system of equations to solve for $n$ and $y$: $$2n + y = 3$$

1. We are given the equation $2n + y = 3$. 2. To express $y$ in terms of $n$, subtract $2n$ from both sides:

1. We are given the system of equations: $$y = 14$$

1. **State the problem:** Simplify the expression $$\frac{3a^5+a^4-3a^3-3a^2+2}{1-a^2}$$. 2. **Factor the denominator:** Note that $$1 - a^2$$ is a difference of squares, which fac

1. The problem is to find the value of $A+B$ given variables $A$ and $B$. 2. Without specific values or expressions for $A$ and $B$, we cannot numerically evaluate $A+B$.

1. **State the problem:** Convert the postfix expression $$A B * C D ^ / E F G + * + H I + J K - / -$$ to its equivalent infix form, then evaluate it using values $$A=100, B=10, C=

1. **Problem 8:** The cost of one eraser and three pencils is 17, and the cost of three erasers and four pencils is 31. Find the cost of 5 erasers. 2. Let the cost of one eraser be

1. The problem is to rewrite the quadratic function $y = x^2 + 4x - 12$ into vertex form. 2. Recall that the vertex form of a quadratic is $y = a(x-h)^2 + k$, where $(h,k)$ is the

1. Stating the problem: Solve the quadratic equation $2x^2 + 3x - 4 = 0$ using the quadratic formula. 2. The quadratic formula to find roots of $ax^2 + bx + c = 0$ is given by:

1. The problem is to simplify the expression $\sqrt{28} + \sqrt{63}$. 2. First, factor the numbers under the square roots to find perfect squares.

1. The problem is to simplify the expression $\sqrt{1000} + \sqrt{90}$. 2. Simplify $\sqrt{1000}$ by factoring out perfect squares:

1. The problem is to simplify the expression $\sqrt{300} - \sqrt{48}$.\n2. Start by simplifying each square root. Break down $300$ and $48$ into their prime factors:\n $$300 = 10

1. We are asked to simplify the expression $\sqrt{8} + \sqrt{2} + \sqrt{72}$.\n2. Start by simplifying each square root where possible:\n - $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt

1. Stating the problem: Simplify the expression $\sqrt{200} - \sqrt{32}$.\n\n2. Simplify each surd by factoring out the perfect squares:\n\n$\sqrt{200} = \sqrt{100 \times 2} = \sqr

1. Stating the problem: Solve the quadratic equation $$3x^2 + x - 2 = 0$$ using the method of completing the square. 2. First, divide the entire equation by 3 to make the coefficie

1. The problem asks to evaluate the expression $$\frac{\sqrt[3]{5} - \sqrt{3}}{\sqrt[3]{5} - \sqrt{3}}$$. 2. Notice that the numerator and denominator are identical.

1. **Problem:** When loaded with bricks, a lorry has a mass of 11,600 kg. The mass of the bricks is three times that of the empty lorry. Find the mass of the bricks. **Step 1:** Le

1. We are given the quadratic equation $3x^2 + x - 2 = 0$. Our goal is to find the roots of this equation. 2. The quadratic equation is in the form $ax^2 + bx + c = 0$ where $a = 3