🧮 algebra

1. The problem is to find the gradients (slopes) of the lines given by the equations j: 4x - y = 8, k: x + 2y - 2 = 0, and l: 4x = 2 + 3y by rearranging them into the form $y = mx

1. **Problem:** Find the equation of the line that passes through the point (6, -7) and is parallel to the line $$2x - 3y + 4 = 0$$. 2. **Formula and rules:**

1. Planteamos el problema: Simplificar las siguientes operaciones con raíces de distintos índices. 2. Recordemos que para multiplicar o dividir raíces del mismo índice, podemos ope

1. **Find the equation of the line that passes through (4,3) with gradient $g=2$.** The formula for the equation of a line with gradient $m$ passing through point $(x_1,y_1)$ is:

1. **State the problem:** We are given a function $f(x)$ and asked to find the values of $x$ for which $f(x) = 0$. 2. **Understand the problem:** The values of $x$ where $f(x) = 0$

1. **Find the equation of the line that passes through (4,3) with gradient 2.** The formula for the equation of a line with gradient $m$ passing through point $(x_1,y_1)$ is:

1. The problem is to find the gradient of a line given by the equation $ax + by + c = 0$. 2. The formula to find the gradient $m$ of a line in the form $ax + by + c = 0$ is:

1. **State the problem:** Solve the equation $5^n = n + 2^n$ for $n$. 2. **Understand the equation:** We want to find the value(s) of $n$ such that the exponential expression $5^n$

1. **State the problem:** Solve the equation $25^{x-1} = 125^{2-x}$ for $x$. 2. **Recall the formula and rules:** We can express both sides with the same base to simplify the equat

1. **State the problem:** Solve the equation $$7^{x+2} - 7^{x+1} + 7^x = 86$$ for $x$. 2. **Rewrite the equation using properties of exponents:** Recall that $$7^{x+2} = 7^x \cdot

1. **State the problem:** We need to find the coordinates where the graph of the function $g(x) = 7x - 7$ intersects the X-axis. 2. **Recall the rule:** The graph intersects the X-

1. **State the problem:** We are given a system of equations: $$x - xy + y = -7$$

1. The problem asks to find which quadratic equation has a discriminant equal to 0. 2. Recall the discriminant formula for a quadratic equation $ax^2 + bx + c = 0$ is:

1. **State the problem:** Simplify the expression $$a^{\log_{a^2} b^4 + \log_a (7b^2)}$$ where $$a, b > 0$$ and $$a \neq 1$$. 2. **Recall logarithm change of base and properties:**

1. დავწეროთ ფუნქცია: $z=\sqrt{4 - x^2 - y^2}$. 2. ფუნქციის განსაზღვრის არე მოიცავს ყველა იმ წერტილს $(x,y)$, სადაც ფესვის შიგთავსი არ არის უარყოფითი, ანუ:

1. Observing the graph, determine if the following statements are true or false. Justify. 1. The function has no y-intercept.

1. **State the problem:** Solve the quadratic equation $$x^2 + 3x - 9 = 0$$. 2. **Formula used:** The quadratic formula to solve $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b

1. **Problem statement:** Find the locus of a point $P(\alpha, \beta)$ such that the line $y = \alpha x + \beta$ is tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

1. **State the problem:** We are given a system of linear equations: $$2x + 3y = 1$$

1. **State the problem:** We are given the quadratic equation $$x^2 - 5x + 4 = 0$$ and its two real solutions $$x_1 < x_2$$. We need to compute the value of $$x_1 + 2x_2$$. 2. **Re

1. **State the problem:** Solve the linear equation $-5x + y = -7$ for $y$ in terms of $x$. 2. **Formula and rules:** To solve for $y$, isolate $y$ on one side of the equation by a