🧮 algebra

1. The problem is to find the value of $y$ in the equation $9^2 + 5^2 = y^2$. 2. This is a Pythagorean theorem problem where $a^2 + b^2 = c^2$ for a right triangle.

1. The problem is to evaluate $\log \frac{1}{2}$.\n\n2. Recall the logarithm rule: $\log \frac{a}{b} = \log a - \log b$.\n\n3. Applying this rule, we get: $$\log \frac{1}{2} = \log

1. **State the problem:** Solve the inequality $$6 \geq -\frac{8}{3} v$$ for $$v$$. 2. **Recall the rule:** When multiplying or dividing both sides of an inequality by a negative n

1. The problem is to solve the system of linear equations: $$6x - 5y = 9$$

1. **State the problem:** We have a number $x$ with a partial prime factorization shown in a tree diagram. We want to find the minimum value of $x$ if $x$ is a perfect cube. 2. **A

1. **State the problem:** Simplify the expression $$\frac{\left(-\frac{2}{5}\right) + \left(\frac{1}{-2}\right)}{\left(\frac{5}{-8}\right) - \left(-\frac{1}{2}\right)}$$. 2. **Rewr

1. **State the problem:** Solve the equation $$5^{-8}(2^x - 1)^3 = 125$$ for $x$. 2. **Rewrite constants:** Note that $125 = 5^3$, so the equation becomes $$5^{-8}(2^x - 1)^3 = 5^3

1. **State the problem:** Solve the equation $$3^{2x - 1} + 1 = 2$$ for $x$. 2. **Isolate the exponential term:** Subtract 1 from both sides:

1. The problem is to solve a quadratic equation by factoring instead of using the quadratic formula. 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$.

1. The problem is to solve the quadratic equation $5x^2 - 11x + 6 = 0$. 2. We use the quadratic formula to find the roots of $ax^2 + bx + c = 0$:

1. **State the problem:** Find the coordinates of point C where line k crosses the x-axis.

1. The problem is to understand the formula for compound interest and how it relates to the given table. 2. The formula for compound interest is:

1. **State the problem:** We have three vegetable patches with areas $w$, $2w + 5$, and $4w$ respectively. The total area is greater than 61 $m^2$. We need to write and solve an in

1. **State the problem:** Samuel thinks of a number $k$. He triples it and then subtracts 11 to get a result less than 43. 2. **Write the inequality:** Tripling $k$ means $3k$. Sub

1. **Problem Statement:** Factor the expression $x^4 - 36$ using the difference of squares formula. 2. **Formula and Rules:** The difference of squares formula is $a^2 - b^2 = (a -

1. **State the problem:** Solve the quadratic equation $x^2 = 2x - 12$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. Stating the problem: Solve the equation $$\frac{4x}{1} \div 2 = x + 7$$. 2. Understand the division by 2: Dividing by 2 is the same as multiplying by $\frac{1}{2}$, so rewrite t

1. **State the problem:** Simplify the expression $$\frac{8}{12} \times x^{-3} \div t^2 \times t^3 \div x^2$$. 2. **Rewrite the expression:**

1. **State the problem:** Simplify the expression $$\frac{8}{12} \times \frac{x^3}{t^2} \times \frac{t^3}{x^2}$$. 2. **Write the expression clearly:**

1. Let's start by stating the problem: You want to understand the rules of logarithms. 2. The logarithm of a number answers the question: to what power must we raise a base number

1. **State the problem:** Factor the expression $$\frac{15x^2}{5xy} + \frac{25x^3}{3xy}$$. 2. **Rewrite each term:** Simplify each fraction by dividing numerator and denominator.