🧮 algebra

1. **Problem statement:** We want to find when the graph of $y=\log_a(x)$ intersects the graph of $y=a^x$ at exactly one point, where $a>1$. 2. **Recall definitions:**

1. Problem a) Simplify $\frac{3}{4} \times 6 - \frac{5}{6} \times 4$ and then divide by 2. 2. Use multiplication of fractions and subtraction:

1. **State the problem:** Solve the linear equation $x + 3y = 15$ for $y$ and understand its graph. 2. **Formula and rules:** To graph a linear equation, it's often easiest to solv

1. The problem is to solve the quadratic equation $x^2 - 2x + 1 = 0$. 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$.

1. **Stating the problem:** Solve the quadratic equation given the system of equations involving $p$ and $q$.

1. The problem asks to evaluate $17 \times (15 \times 16)$ and determine which expression among the choices A, B, and C is equivalent to it. 2. First, calculate the value inside th

1. **Stating the problem:** Plot the graphs of the linear equations: a) $2x - y = -4$

1. **Problem statement:** We have a cable curve described by a quadratic function $f(x) = a x^2 + b x + c$ between two supports A and B.

1. The problem is to simplify the expression $$4.2a + 3b - 0.5a$$ and find the coefficient of $$a$$ in the simplified form. 2. To simplify, combine like terms. Like terms are terms

1. **State the problem:** Simplify the expression $$\frac{5}{6}b + 5 - \frac{2}{3}b - 2$$ and express it in the form $$\frac{1}{6}b + [?]$$ where [?] is a number to find. 2. **Comb

1. **State the problem:** Solve the linear equation $$11 = 3x - 4$$ for $$x$$. 2. **Add 4 to both sides** to isolate the term with $$x$$:

1. **State the problem:** We need to find the value of the expression $$56 \div y^3 - 2 \cdot 3$$ when $$y = 2$$. 2. **Write the expression clearly:** $$56 \div y^3 - 2 \cdot 3$$.

1. **State the problem:** Fully factorise the expression $6x + 18y + 24z$. 2. **Identify the greatest common factor (GCF):** Look for the largest number that divides all coefficien

1. The problem is to evaluate $10$ raised to the power of $456$. 2. The expression is written as $$10^{456}$$ which means multiplying $10$ by itself $456$ times.

1. The problem is to solve the equation $$\frac{2x+3}{x-1} = 4.$$\n\n2. We use the property that if $$\frac{a}{b} = c$$ and $$b \neq 0$$, then $$a = bc$$. Here, $$a = 2x+3$$, $$b =

1. **State the problem:** Find the equations of two lines passing through the point $(-4, 2)$ whose perpendicular distance from the origin is 2. 2. **Recall the distance formula fr

1. **State the problem:** Solve the equation $$-5 + 2 \ln(3x) = 5$$ for $x$, rounding to the nearest hundredth. 2. **Isolate the logarithmic term:** Add 5 to both sides:

1. مسئله: محاسبه مقدار عبارت $ (x - y + y)(x + y - y) $ است. 2. ابتدا عبارت را ساده می‌کنیم. توجه کنید که در هر پرانتز، $-y + y = 0$ و $+ y - y = 0$ است.

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. The problem asks to describe the end behavior of the polynomial function $$f(x) = 5x - 2$$. 2. The end behavior of a polynomial function depends on the leading term. Here, the l

1. **State the problem:** We need to solve the equation $$\frac{2x+4}{3} = 5$$ for $x$. 2. **Formula and rules:** To solve equations involving fractions, multiply both sides by the