🧮 algebra

1. **State the problem:** Solve the system of linear equations using Gauss-Jordan elimination: $$\begin{cases}-0.3x - 0.8y = -9 \\ 0.6x - 1.4y = -15 \end{cases}$$

1. **State the problem:** We need to find how many adults and children attended the movie given that 61 people attended in total and the total amount paid was 472. 2. **Define vari

1. **State the problem:** A street vendor has 360 T-shirts total, some short sleeve and some long sleeve. Short sleeve shirts sell for 14 each, long sleeve for 16 each. Total sales

1. The problem asks to evaluate the expression $2x - y$ given the values $x=9$ and $y=-14$. 2. The formula is simply the expression itself: $$2x - y$$

1. **State the problem:** Simplify the expression $$-2x^4y\sqrt{16y^{13}} - 2^{3}\sqrt{16x^{12}y^{16}} - x^{2}y^{3}\sqrt[3]{-54x^{6}y^{7}}$$ 2. **Recall the rules:**

1. **State the problem:** Solve the equation $$-0.8(10 - x) = 36$$ for $x$. 2. **Use the distributive property:** Multiply $-0.8$ by each term inside the parentheses.

1. **State the problem:** Expand and simplify the expression $\frac{1}{2}(T - 4)^2$. 2. **Recall the formula:** The square of a binomial $(a - b)^2$ is expanded as $$ (a - b)^2 = a

1. **State the problem:** Solve the equation $x^2 + \frac{3}{2}x = 0$. 2. **Formula and rules:** To solve quadratic equations, we can factorize or use the quadratic formula. Here,

1. **State the problem:** Solve the equation $$\frac{x}{9} = \frac{5}{x}$$ for $x$. 2. **Use the cross-multiplication rule:** When two fractions are equal, their cross products are

1. State the problem. Solve for $w$ in the equation $5w + 9 = 39$.

1. The problem is to graph the linear inequality $y \geq 10 - x$. 2. First, we graph the boundary line $y = 10 - x$. This is a straight line with slope $-1$ and y-intercept $10$.

1. Problem: Kürzen Sie den Bruchterm $\frac{3ab}{6a}$ soweit wie möglich. 2. Formel: Um Brüche zu kürzen, teilt man Zähler und Nenner durch ihren größten gemeinsamen Teiler (ggT).

1. **State the problem:** Solve the factoring problem $$3x^2 - 18x = -27$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. The problem asks to identify the vertex, focus, directrix, and axis of symmetry for the parabola given by the equation $$y = -3$$ and related information. 2. Since $$y = -3$$ is

1. **State the problem:** We need to solve the equation $$\frac{2x+4}{x-1} = 3$$ for $x$. 2. **Recall the formula and rules:** To solve a rational equation like this, multiply both

1. **State the problem:** Solve the quadratic equation $$7r^2 - 14r = -7$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. The problem appears to be a set of fractions: $\frac{9}{4}$, $\frac{11}{2}$, $\frac{11}{3}$, and $\frac{11}{4}$. We will simplify or compare these fractions. 2. To compare or si

1. **State the problem:** We need to find the input values $x$ such that $t(x) = 12$ and $t(x) = 17$ for the function $$t(x) = 5x^2 - 3x + 4.$$ 2. **Use the formula:** Set the func

1. **State the problem:** Solve the equation $$2 = 3^{0.01t}$$ for $t$. 2. **Recall the formula and rules:** To solve for $t$ when it is in the exponent, use logarithms. The key pr

1. The problem is to find the values of $x$ for which $r(x) = 9.3$ and $r(x) = 20$ given the function $r(x) = 5 \ln(1.4)(1.4^x)$. 2. The function is given as $r(x) = 5 \ln(1.4)(1.4

1. **State the problem:** Factor the polynomial $$4x^3 - x^2 + 8x - 2$$ by grouping. 2. **Group terms:** Group the polynomial into two pairs: