🧮 algebra

1. **State the problem:** Solve the system of equations using substitution: $$4x + 3y = 37$$

1. **State the problem:** Factor the expression $$x^2 + 2x + 1 - y^2$$. 2. **Recognize the structure:** The expression can be seen as a difference of squares because $$x^2 + 2x + 1

1. **State the problem:** We are given that the area of Manitoba is about $1 \frac{1}{5}$ times the total area of the four Atlantic provinces, and the area of Yukon Territory is ab

1. **State the problem:** Calculate the product of $2 \frac{3}{4}$ and $-4 \frac{2}{3}$. 2. **Convert mixed numbers to improper fractions:**

1. **State the problem:** Find the zeros (roots) of the quadratic equation $$5x^2 + 19x + 12 = 0$$. 2. **Formula and approach:** To find the roots, we can factor the quadratic if p

1. **State the problem:** Evaluate the expression $$4x^2 - by^2 - by$$ for $$x=2$$, $$b=\frac{1}{2}$$, and $$y=-3$$. 2. **Write the expression with substituted values:**

1. The first problem asks to sketch the function $f(x) = (x - 5)^2$. 2. This is a parabola that opens upwards with its vertex at the point $(5, 0)$.

1. **Problem statement:** We want to find the correct equation for the rocket's weight $W$ at time $t$ where $0 \leq t \leq T$, given initial weight $W_1$ and final weight $W_T$ at

1. **State the problem:** The ratio of female to male employees is initially 2:5. After hiring 6 more women, the ratio becomes 7:10. We need to find the total number of employees a

1. **State the problem:** We have a hospital with two wards, A and B. Ward A has 40% of beds and 60% of turnover of 120 million. Ward B has 60% of beds and 40% of turnover. The bed

1. **Problem statement:** Claudia spends $x$ percent of her monthly work time on the project "Customer Survey" for the first $a$ months, and $y$ percent for the remaining $b$ month

1. **State the problem:** 8 employees complete the inventory in 6 days. We want to find how long it takes if 4 more employees join, making 12 employees. 2. **Formula used:** Work d

1. **State the problem:** We have a box formed by cutting out squares of side $x$ cm from each corner of a 9 cm by 9 cm square sheet and folding up the sides. We want to express th

1. **State the problem:** We have a box formed by cutting out squares of side $x$ cm from each corner of a $9 \times 9$ cm square sheet and folding up the sides. We want to express

1. **State the problem:** Find the equation of the line passing through the points $(-3, h(-3))$ and $(1, h(1))$ where $h(x) = x^3 + 2$. 2. **Calculate the points:**

1. **State the problem:** Solve for $a$ in the equation $2a + 0.8\overline{3} = 4.6\overline{6}$. 2. **Understand repeating decimals:**

1. **State the problem:** Find the equation of the secant line passing through the points $(-3, g(-3))$ and $(5, g(5))$ on the curve defined by $g(x) = x^2 - 9x$. 2. **Find the coo

1. **State the problem:** Solve for $x$ in the equation $$\frac{8x}{3} = \frac{x}{5} - 74$$. 2. **Write down the equation:** $$\frac{8x}{3} = \frac{x}{5} - 74$$.

1. The problem is to solve the equation $x^2 - 5x + 6 = 0$. 2. We use the quadratic equation formula or factorization method. Here, factorization is simpler.

1. **Stating the problem:** We need to expand the algebraic expressions given: (e) $ (5 - 8x)^2 $

1. **State the problem:** We need to find the value of the expression $$\frac{7 - 3x}{2}$$ when $$x = -9$$. 2. **Substitute the value of $$x$$:** Replace $$x$$ with $$-9$$ in the e