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15 problems match “systems of linear equations”. Clear

Red 4-door cars in a parking lot ●○○○○ 10 ptsThere are 20 cars in a parking lot. All of the cars are red or white, and all of the cars are either 2-door or 4-door. Of these cars, 12 are red, 15 are 4-door, and 4 are 2-door and white. How many of the cars are 4-door and red?
Sum of three numbers from pairwise sums ●○○○○ 10 ptsThree numbers are added two at a time, and the three sums are 29, 46, and 53. Find the sum of all three numbers.
Three numbers from pairwise sums ●○○○○ 10 ptsLet the three numbers be $x$ , $y$ and $z$ with $x+y=10$ , $y+z=19$ and $x+z=21$ . Find $x$ , $y$ and $z$ .
Coordinates of a vector in R^4 ●○○○○ 10 ptsLet $a_1 = [1, 1, 0, 0]$ , $a_2 = [0, 1, 1, 0]$ and $a_3 = [0, 0, 1, 1]$ . Find the coefficients $\alpha_1, \alpha_2, \alpha_3$ such that $v = [1, 3, 5, 3] = \alpha_1 a_1 + \alpha_2 a_2 + \alpha_3 a_3$ , that is, find the coordinate representation $[\alpha_1, \alpha_2, \alpha_3]$ of $v$ .
Largest of four integers from triple averages ●●○○○ 20 ptsDewa writes down a list of four integers. He calculates the average of each group of three of the four integers. These averages are $32, 39, 40, 44$ . What is the largest of the four integers?
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?
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).
Trash can unit prices and minimum type A quantity ●●○○○ 20 ptsA community buys two models of trash cans, A and B. Buying 3 type-A and 4 type-B cans costs a total of $580, and buying 6 type-A and 5 type-B cans costs a total of $860. (1) Find the unit price of a type-A can and of a type-B can. (2) A total of 200 cans of types A and B are needed, and the total cost must not exceed $15000. What is the least number of type-A cans that should be purchased?
Coefficients of a vector in a basis ●●○○○ 20 ptsGiven vectors $\vec{a} = (1, -1)$ , $\vec{b} = (2, 1)$ and $\vec{c} = (-2, 1)$ , suppose $\vec{c} = x\vec{a} + y\vec{b}$ with $x, y \in \mathbb{R}$ . Find $x - y$ .
Decomposing a vector in a basis ●●○○○ 20 ptsWrite the vector $x=(-5, -5, 5)$ as a linear combination of $p=(-2, 0, 1)$ , $q=(1, 3, -1)$ and $r=(0, 4, 1)$ , that is, find $\alpha$ , $\beta$ , $\gamma$ with $x=\alpha p+\beta q+\gamma r$ .