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57 problems match “linear equations”. Clear

Two numbers with a percent relation ●●○○○ 20 ptsThe sum of two numbers is 4000. $15\%$ of one number equals $25\%$ of the other number. Find the two numbers.
Sum of alternate terms of a sequence ●●●○○ 50 ptsA sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?
Integer point in a composed functional equation ●●●○○ 50 ptsThe function $f(x)$ satisfies $f(f(x)) = 6x - 2005$ for all real numbers $x$ . There exists an integer $n$ such that $f(n) = 6n - 2005$ . Find $n$ .
Blue rectangles on a blackboard ●●●○○ 50 ptsBlue and red rectangles are drawn on a blackboard. Exactly 7 of the rectangles are squares. There are 3 more red rectangles than blue squares. There are 2 more red squares than blue rectangles. How many blue rectangles are there?
Parameter values with no solution ●●●○○ 50 ptsDetermine all real numbers $a$ for which the equation $\frac{a}{1-\frac{1}{1+\frac{1}{x}}}=2023 x$ has no solutions. Enter the values in increasing order.
Sum of entries of inverse matrices ●●●○○ 50 ptsThe matrices $\begin{pmatrix} a & 1 & b \\ 2 & 2 & 3 \\ c & 5 & d \end{pmatrix}$ and $\begin{pmatrix} -5 & e & -11 \\ f & -13 & g \\ 2 & h & 4 \end{pmatrix}$ are inverses of each other. Find $a + b + c + d + e + f + g + h$ .
Piecewise function with equal values ●●●○○ 50 ptsLet $a \neq 0$ be a real number, and define the function $f(x)$ by $f(x)=\begin{cases}2x+a, & x<1,\\ -x-2a, & x\geq 1.\end{cases}$ If $f(1-a)=f(1+a)$ , find the value of $a$ . Give your answer as a decimal.