Master Algebra 1 Word Problems with our Comprehensive Practice

I want to know how can i linear programming be used in business

Write a equation in point -slope form for the line with the given slope and through the given point (-2,-6) and has slope 8/3

The cost of a school banquet is $70 plus $15 per person attending. Determine the linear equation that models this problem. What is the cost for 44 people?

Write an nth term rule for 6,−30,150,−750,... and find ${\displaystyle a_6}$

Find the integers if the product of two numbers is an integers

Write an equation in point-slope form for the given (6,8) and is parallel to the line given by y=-5x+4

Let $x,y,z>0$ and such
$$\{\begin{array}{l}{\displaystyle \frac{x}{xy-<!--\; -\; -->z^2}}=-<!--\; -\; -->{\displaystyle \frac{1}{7}}\\ {\displaystyle \frac{y}{yz-<!--\; -\; -->x^2}}={\displaystyle \frac{2}{5}}\\ {\displaystyle \frac{z}{zx-<!--\; -\; -->y^2}}=-<!--\; -\; -->3\end{array}$$
show that:
$$x+y+z=6$$

Find the eighth term of the geometric sequence whose first three terms are 3, 6 and 12

Write an equation of a line passing through (-8, -5), perpendicular to x+4y=6

The sum of two numbers is 15. One number is twice as large as the second number. What are the two numbers? Which let statement would you use?

Please tell whether the sequence 1,3,7,15,31,... is geometric

Local times and quadratic variation
Suppose we have a continuous local martingale $(M_t{)}_{t\ge <!--\; \ge \; -->0}$, where $M_0=0$. Let $L_t^0$ denote the local time of this process at zero. I would like to prove the simple statement that:
$$inf\{t:L_t^0>0\}=inf\{t:⟨<!--\; ⟨\; -->M{⟩<!--\; ⟩\; -->}_t>0\},$$
where the bracket denotes the quadratic variation process.
My initial idea was to use the occupation density formula, which relates local times and the quadratic variation through the integrals involving non-negative Borel measurable functions. However, the formula integrates against the superscript of L rather than the subscript (that is, one measures the change with respect to the position of the process, not with respect to the time for a fixed position). Therefore, I do not hope that this could be useful.
I am not aware of any other (useful) results connecting the quadratic variation and the local times, so I would appreciate any help. I appreciate the fact that (heuristically) the local time measures the time a process spends at a particular point, with the clock given by the quadratic variation rather than the time indexing the process, but I can't quite get a rigorous proof of this.

Is this a geometric sequence 8,4,2,1,1/2,1/4?

Write a rule for the nth term of the geometric sequence and then find ${\displaystyle a_5}$ given ${\displaystyle a_3=-144,r=0.5}$

Write an equation of a line through (5,-1) parallel to ${\displaystyle y=\frac{1}{9}x+1}$

How would you best describe the rate of growth of the function $f(x)=cx{r}^x$ ?
Consider the function $f(x)=cx{r}^x$ , where both r and c are constants and we have cases: (a) r<1, (b) r>1. Regarding terminology, how would you best describe the asymptotic growth of f(x) in cases (a) and (b)? Though you could state that exponential growth and decay will dominate the asymptotics of the function, strictly speaking it would be incorrect to call (a) exponential growth and (b) exponential decay, right?

Find the slope of any line perpendicular to the line passing through (8,12) and (5,−2)

Find the slope that is perpendicular to the line (4, 19) and (-3, 5)

Find the fifth term in a geometric sequence in which the fourth term is 5, the sixth term is 7, and the common ratio is negative

Find the sum of the infinite geometric series, a(1)= -5, r= 1/6