🧮 algebra

1. **State the problem:** Expand and simplify the expression $5n + 3(4 + 2n)$. 2. **Use the distributive property:** Multiply $3$ by each term inside the parentheses:

1. **Problem:** Solve the equation $x^2 = \frac{16}{169}$ using roots. 2. **Formula and rule:** To solve $x^2 = a$, use the square root as the inverse operation: $$x = \pm \sqrt{a}

1. Нека разгледаме първата система от уравнения: $$\begin{cases} x + y = 3 \\ x^2 + y^2 - x = 3 \end{cases}$$

1. **Stating the problem:** Simplify the expression $$7^5 \cdot 2^5 \times \frac{1}{14^5} \times 14^{10} \times 14^5$$. 2. **Recall the properties of exponents:**

1. The problem is to simplify the expression $5^4 \cdot 5$. 2. Recall the rule for multiplying powers with the same base: $a^m \cdot a^n = a^{m+n}$.

1. **State the problem:** We have the simultaneous equations: $$ax + by = a + b$$

1. **State the problem:** Solve the equation $$y - 2 = \frac{y^2}{9}$$ for $y$. 2. **Rewrite the equation:** Multiply both sides by 9 to eliminate the denominator:

1. **State the problem:** Solve for $k$ in the equation $$3(k + 1) + 1 = 7$$. 2. **Apply the distributive property:** Multiply 3 by each term inside the parentheses.

1. **State the problem:** Solve for $q$ in the equation $$-q - 12q + 12q - 8 = 12$$. 2. **Combine like terms:** The terms with $q$ are $-q$, $-12q$, and $+12q$. Combine them:

1. **State the problem:** Given the ratio $\frac{a}{b} = \frac{3}{5}$ and $a = 2$, find the value of $b$. 2. **Formula used:** For ratios, if $\frac{a}{b} = \frac{3}{5}$, then $a$

1. **State the problem:** Kelsey knitted a scarf with green and blue stripes. The ratio of green to blue stripes is 4 to 5. We know she knitted 48 rows of green stripes and want to

1. **State the problem:** Solve the quadratic equation $$x^2 - |a-1| x - 1 = 0$$ for $x$. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions

1. The problem is to solve the inequality $$670 - 37 \leq 78$$ and express the solution in interval form with two decimal places. 2. First, simplify the left side:

1. **State the problem:** Solve the quadratic inequality $$-x^2 + 2x - 10 \geq 0$$. 2. **Rewrite the inequality:** It is often easier to analyze when the quadratic term is positive

1. **State the problem:** Find the equation of the line in point-slope form passing through the points $(9, -\frac{1}{2})$ and $(7, \frac{3}{2})$. 2. **Recall the formula:** The po

1. **State the problem:** Find the equation of the line passing through points $(-9,7)$ and $(9,1)$ in point-slope form and slope-intercept form. 2. **Find the slope $m$:** Use the

1. **Stating the problem:** Simplify the expression $$\frac{-22-4xh}{14-2xh} \div \frac{22-14xh}{x^2-8x-28} \div \frac{x^2-6x}{x^2-6x}.$$ 2. **Rewrite the division as multiplicatio

1. El problema es resolver la ecuación dada (aunque no se especifica, asumiremos que se refiere a una ecuación algebraica común). 2. Para resolver una ecuación, debemos aislar la v

1. El problema es resolver la ecuación $-x + y = 9$ dado que $x = -15$ y $y = 6$. 2. La ecuación es una igualdad que relaciona $x$ y $y$. Para verificar si los valores dados satisf

1. Stating the problem: Simplify the expression \(\frac{-22 - 4xh}{14 - 2xh} \div \frac{22 - 14xh}{x^2 - 8x - 28} \div \frac{x^2 - 6x}{x^2 - 6x}\). 2. Rewrite the expression as mul

1. **State the problem:** Solve the simultaneous equations: $$y = -x + 2$$