🧮 algebra

1. The problem is to analyze the system of linear equations from set 7: $$y = -\frac{1}{2}x - 2$$

1. The problem asks which logarithm property is demonstrated by the equation $5^{109-5-10939} = 1098$. 2. Let's clarify the properties of logarithms:

1. The problem asks which logarithm property is demonstrated by the equation $$109g8 + 1096 = 109g48$$. 2. First, let's rewrite the equation in a clearer form assuming the notation

1. **State the problem:** Prove by mathematical induction that for all positive integers $n$, $$1 + 4 + 4^2 + \cdots + 4^n = \frac{1}{3}(4^{n+1} - 1)$$

1. **Problem statement:** Given the sequence 5, 11, 17, 23, 29, find: (a) the expression for the nth term,

1. **State the problem:** Factor out the greatest common factor (GCF) from the expression $$18m^6 - 12m^2$$. 2. **Identify the GCF:**

1. **Solve the first system of inequalities:** Given:

1. **Stating the problem:** We have the charge formula for a repair:

1. **State the problem:** Solve the equation $19l - 57 = 19$ for $l$. 2. **Add 57 to both sides** to isolate the term with $l$:

1. The problem asks to find the value of $Q$, which represents the number of hours Khan worked overtime at double time. 2. To solve this, we need the formula relating total income,

1. **State the problem:** We need to find Anna's weekly gross pay $P$ given her hourly rates and hours worked, including overtime. 2. **Understand the pay rates:**

1. **State the problem:** Lal's normal rate of pay is 10 per hour. We need to find his new hourly rate after an 8% pay rise. 2. **Formula used:** To calculate the new pay rate afte

1. **State the problem:** Calculate Samu's gross pay for last week given different hourly rates for weekdays, Saturday, and Sunday, and his working hours with unpaid breaks. 2. **H

1. Problem: Odrediti Lagranžov interpolacioni polinom za funkciju $f(x) = 1 - 3^x$ sa čvornim tačkama $x_i = i$, gde je $i = 0, 1, 2, 3$. 2. Prvo izračunavamo vrednosti funkcije u

1. **State the problem:** We are given two equations based on the cost of rackets and squash balls: - The cost of 2 rackets and 3 squash balls is 21.63.

1. The problem states that the ratio of water used for bathing, washing laundry, and washing the car is 9 : 7 : 1. 2. We know the amount of water used for bathing is 144 liters.

1. **State the problem:** We need to solve for $D$ where $$D = 2 \times \sqrt{\frac{3}{2}} - 3 \times \sqrt{\frac{2}{3}} - 1.$$\n\n2. **Recall the properties of square roots and si

1. **Stating the problem:** Simplify the expression $$\frac{16}{35} \times 0.25 \times 35$$. 2. **Convert decimals to fractions:** Note that $$0.25 = \frac{1}{4}$$.

1. **State the problem:** A company gave a total of 15500 to three charities: Charity A received 4000, Charity B received 5500, and Charity C received the remaining amount. We need

1. **State the problem:** Sandy bought three items: a calculator, a thumb drive, and a phone case. The ratio of their costs is 7 : 13 : 4, and the total cost is 72. 2. **Formula an

1. **State the problem:** We have a function $g$ defined on the set $\{-2, 1, 3, 5\}$ with the rule $g(x) = -2x + 1$. We want to find the values of $g(x)$ for each $x$ in the domai