Master Algebra 1 Word Problems with our Comprehensive Practice

A cell phone distributor earns a yearly salary of $28,000 plus a 17.5% commission on sales. How do you find the total earnings for a year when the sales are $38,000?

Convert a non-convex QCQP into a convex counterpart
We consider a possibly non-convex QCQP, with nonnegative variable $x\in <!--\; \in \; -->{\mathbb{R}}^n$,
$$\begin{array}{rlrl}& \underset{x}{\text{minimize}}& & f_0(x)\\ & \text{subject to}& & f_i(x)\le <!--\; \le \; -->0,i=1,\dots <!--\; \dots \; -->,m\\ & & & x\ge <!--\; \ge \; -->0\end{array}$$
where $f_i(x)=\frac{1}{2}{x}^T{P}_{i}x+q_i^{T}x+r_i$, with $P_i\in <!--\; \in \; -->{\mathbb{S}}^n$, $q_i\in <!--\; \in \; -->{\mathbb{R}}^n$, and $r_i\in <!--\; \in \; -->\mathbb{R}$, for $i=0,\cdots <!--\; \cdots \; -->,m$. We do not assume that $P_i\ge <!--\; \ge \; -->0$, so this need not to be a convex problem.
Suppose that $q_i\le <!--\; \le \; -->0$ and $P_i$ have non-positive off-diagonal entries. Explain how to reformulate this problem as a convex problem.
What I Have Done
Since both objective function and constraints are possibly non-convex, I consider their common structure, that is $f_i(x)=\frac{1}{2}{x}^T{P}_{i}x+q_i^{T}x+r_i,i=0,1,2,\cdots <!--\; \cdots \; -->,m$. In this structure, $q_i^{T}x+r_i$ part is affine, so it does not cause problem. Then I only consider $x^T{P}_{i}x$.
The first thing that arises in my mind is to decompose $P_i$ into certain form and make $x^T{P}_{i}x$ become something like $y^{T}Qy$ where $Q\ge <!--\; \ge \; -->0$. However, I did not find anything useful (actually it seems that I should not or any non-convex quadratic programming could be solved in this way).
Even though the previous random thought does not seem to work, I still think certain transformation is necessary, and perhaps some non-linear transformation $y_i=\mathrm{\varphi <!--\; \varphi \; -->}(x_i)$. But coming up with this may need some "magic" and up to now have found anything that helps.

Find the 7th term of the geometric sequence where a1 = –4 and a5 = –1,024

What is the sum of the arithmetic sequence 4, 11, 18 …, if there are 26 terms?

Need to solve the following product:
$$\underset{k=1}{\overset{N}{\prod <!--\; \prod \; -->}}{\mathbb{1}}_{\{y_k\ge <!--\; \ge \; -->1\}}$$
Is there a way to simplify it or write it in a closed-form expression?

Define the parabola $y=a{x}^2+bx+c$ such that $a,b,c$are positive integers. Suppose that the roots of the quadratic equation $a{x}^2+bx+c=0$ are $x_1,x_2$ such that $|x_1|,|x_2|>1$, then compute the minimum value of $abc$ and when the minimum occur what is the value of $a+b+c$.

Find the explicit formula for a geometric sequence 9,19, 39,79,159,...

Write an equation of a line through (4,-1) and parallel to y= 2x +1

Find a1 in the arithmetic series with s7=287 and d=12

How to check if the inequality $Ax<b$ admits at least one solution. Entries of $A$, $x$ and $b$ are taken in $\mathbb{Z}$

Find the smaller if 3 times the smaller of 2 consecutive integers is added to 4 times the larger, the result is 102

Write the first six terms of the sequence ${\displaystyle a_n=\frac{n+2}{2n}}$

Find the equation of the line that passes through (−2,7) and is perpendicular to the line that passes through the following points: (1,3),(−5,6)

William opens a credit card with an APR of 18.63% compounded monthly. How much is charged in interest this month if his balance is $835?

Find the point-slope form of the equation of the line passing through the Point: (1, –7); Slope: -2/3

Use the quadratic formula to solve the equation.
$$x^2-<!--\; -\; -->4x-<!--\; -\; -->12=0$$

Since every function can be divided in a even and a odd part:
$$f_e(x)=\frac{f(x)+f(-<!--\; -\; -->x)}{2}$$
$$f_o(x)=\frac{f(x)-<!--\; -\; -->f(-<!--\; -\; -->x)}{2}$$
$$f(x)=f_e(x)+f_o(x)$$
How can I obtain the function by it's even part?

We have been going through how to solve the system of equations known as Ax=b, where A is a matrix, x is a vector and b is a vector. I understand that if we have A and b we must find out what x is, this happens via Gauss-Jordan elimination, back substitution, etc. What does solving the linear system of equations actually mean though?

In eight years, peter will be three times old as he was eight years ago, how old is peter now?

Determine whether each sequence is an arithmetic sequence: 4, 9, 14, 19...