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57 problems match “linear equations”. Clear

Crossing the horizontal asymptote ●●○○○ 20 ptsFor which value of $x$ does the function $f(x) = \frac{2x^2 - 5x - 7}{x^2 - 4x + 1}$ cross its horizontal asymptote?
Equation with square and cube roots ●●○○○ 20 ptsSolve the equation $\frac{\sqrt{x}+\sqrt[3]{x}}{\sqrt{x}-\sqrt[3]{x}}=3$ for real $x$ .
Median of consecutive integer sets ●●○○○ 20 ptsLet $a$ , $b$ , and $c$ be integers with $a < b < c$ . Let $S$ be the set of all integers from $a$ to $b$ , inclusive, and $Q$ the set of all integers from $b$ to $c$ , inclusive. The median of $S$ is $\frac{3}{4}b$ and the median of $Q$ is $\frac{7}{8}c$ . Let $R$ be the set of all integers from $a$ to $c$ , inclusive. What fraction of $c$ is the median of $R$ ? Give your answer as a decimal.
Sum of initial candle lengths ●●○○○ 20 ptsTwo candles have lengths that differ by 32 cm. The longer one is lit at 15:00 and the shorter one at 19:00. At 21:00 they have the same length. The longer one burns out completely at 22:00, and the shorter one at midnight. Each candle burns at a constant rate. What is the sum of the initial lengths of the candles, in centimetres?
Boards in a simultaneous chess exhibition ●●○○○ 20 ptsA master gives a simultaneous chess exhibition on several boards, with one game played on each board. In the first two hours, he won $10\%$ of the total number of games, and 8 opponents drew their games with him. Over the next two hours, he played the remaining opponents: he won $10\%$ of those games, lost two games, and drew the remaining 7 games. On how many boards was the exhibition played?
Students who like to dance ●●○○○ 20 ptsOf the students attending a school party, $60\%$ are girls, and $40\%$ of the students like to dance. After these students are joined by $20$ more boys, all of whom like to dance, the party is now $58\%$ girls. How many students at the party now like to dance?
Share of total area from rose fractions ●●○○○ 20 ptsTwo fields are planted with roses and lavender. $65\%$ of the area of the first field is roses, $45\%$ of the area of the second field is roses, and $53\%$ of the area of both fields together is roses. What percentage of the total area (of both fields together) is the area of the first field?
Triangle sides from heights ●●○○○ 20 ptsIn triangle $ABC$ , the height to side $a$ is $6$ cm and the height to side $b$ is $9$ cm. Side $a$ is $4$ cm longer than side $b$ . Calculate the side lengths $a$ and $b$ (in cm).
Scout troop after percentage growth ●●○○○ 20 ptsLast year, there were 30 more boys than girls in a scout troop. This year, the number of children in the troop increased by $10\%$ , the number of boys increased by $5\%$ , and the number of girls increased by $20\%$ . How many children are in the troop this year?
Four-element set from its triple sums ●●○○○ 20 ptsLet $A=\{a_1, a_2, a_3, a_4\}$ be a set of four numbers. The set $B=\{-1, 3, 5, 8\}$ consists of the sums of the elements of all the three-element subsets of $A$ . Find the four elements of $A$ , in increasing order.