🧮 algebra

1. **State the problem:** We need to find an angle whose complement is 5 times as large as the angle itself. 2. **Define variables:** Let $x$ be the measure of the angle in degrees

1. **State the problem:** We are given that line \(\ell\) bisects segment \(KP\) at point \(L\). This means \(L\) is the midpoint of \(KP\), so \(KL = LP\).

1. **Problem statement:** Find the equation of lines with given slopes and describe the meaning of $m$ and $b$ in $y=mx+b$. Also, describe and graph $y=-3x+6$ without technology. 2

1. **State the problem:** Factor the expression $x^2 - 9$. 2. **Recall the formula:** This is a difference of squares, which factors as $$a^2 - b^2 = (a - b)(a + b)$$ where $a = x$

1. Stating the problem: factor $x^2-9$. 2. Use the difference of squares formula: $$a^2-b^2=(a-b)(a+b)$$

1. The problem is to evaluate the binomial coefficient $\binom{2}{3}$. 2. The binomial coefficient $\binom{n}{k}$ is defined as $\frac{n!}{k!(n-k)!}$ where $n!$ is the factorial of

1. **State the problem.** We need to choose the equation of a line that crosses the y-axis at $-2$ and the x-axis at $5$.

1. **State the problem:** Solve the linear equation $16 - 2t = 5t + 9$ for $t$. 2. **Write down the formula and rules:** To solve for $t$, we need to isolate $t$ on one side of the

1. **State the problem:** Simplify the expression $$\frac{\frac{1}{x} - \frac{1}{y}}{y^2 - x^2}$$ and identify which of the given options (a to e) it equals. 2. **Rewrite the numer

1. **State the problem:** Simplify the expression $$\frac{x}{x+2} - \frac{7}{x-2}$$ and identify which option (a, b, c, d, or e) matches the simplified form. 2. **Formula and rules

1. **State the problem:** Solve the equation $$\frac{x}{x+2} - \frac{7}{x-2} = 0$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, find a common denominator a

1. **State the problem:** Simplify the expression $$\frac{2}{x+2} - \frac{7}{x-2}$$ and compare it with the given options. 2. **Find a common denominator:** The denominators are $x

1. **State the problem:** Simplify the expression $$-2x(x+3) - (x+1)(x-2)$$ and identify which option matches the result. 2. **Apply distributive property:**

1. **State the problem:** Simplify the expression $$\frac{x^2 - 25}{x^2 + 5x} \div \frac{xy + 6x - 5y - 30}{5x - 15}$$. 2. **Rewrite division as multiplication by the reciprocal:**

1. **Problem:** Which equation matches the line shown on the graph? 2. **Use the line form:** $y=mx+b$.

1. **Problem:** Which equation has the same solution as $5x + 8 = x - 9$? 2. **Use the equation-solving steps:**

1. **State the problem:** We need to find which equation among the options has the same solution as the equation $$5x + 8 = x - 9$$. 2. **Solve the original equation:**

1. **State the problem:** We need to find which equation among the options has the same solution as the equation $$5x + 8 = x - 9$$. 2. **Solve the original equation:**

1. **Problem:** Simplify $$\frac{x^2-25}{x^2+5x}\div\frac{xy+6x-5y-30}{5x-15}$$

1. **State the problem:** Simplify the expression $$\frac{x^2 - 25}{x^2 + 5x} \div \frac{xy + 6x - 5y - 30}{5x - 15}$$. 2. **Rewrite division as multiplication by reciprocal:**

1. **Planteamiento del problema:** Tenemos la función por partes $$g(x) = \begin{cases} x + 2 & \text{si } x < -1 \\ x^2 - 3 & \text{si } x \geq -1 \end{cases}$$ Se pide calcular: