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57 problems match “linear equations”. Clear

Evaluate a function from two values ●●○○○ 20 ptsIf $f(1)=5$ , $f(2)=8$ and $f(x)=ax+bx+2$ , what is the value of $f(3)$ ?
Complex equation with a conjugate ●●○○○ 20 ptsLet $z$ be a complex number and let $z'$ denote its complex conjugate. Find $z$ if $\frac{z}{2}+3=\frac{z'}{3}-2$ .
First term of a recurrence with a constant subtraction ●●○○○ 20 ptsA sequence $a_1, a_2, a_3, \ldots$ satisfies $a_n = 3a_{n-1} - x$ for every $n \ge 2$ , where $x$ is an integer that is the same for all terms. Given that $a_2 = 7$ and $a_4 = 43$ , find $a_1$ .
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?
Input value of a linear function ●●○○○ 20 ptsGiven that $f\left(\frac{x}{2} - 1\right) = 2x + 3$ for all $x$ , and $f(m) = 6$ , find the value of $m$ . Give your answer as a decimal.
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 $x=\alpha p+\beta q+\gamma r$ .
Orthocentre of a triangle ●●○○○ 20 ptsFind the coordinates of the orthocentre of the triangle with vertices $A(5,6)$ , $B(2,4)$ and $C(1,-3)$ . Round each coordinate to two decimal places.
Two equations with percents ●●○○○ 20 ptsThe sum of $x$ and $y$ is 150. The sum of 5 percent of $x$ and 10 percent of $y$ is 11. Find $x$ and $y$ .
Coordinates in a non-standard basis ●●○○○ 20 ptsLet $v = [6, 3, 2, 5]$ , $a_1 = [2, 2, 2, 3]$ , $a_2 = [1, 0, -1, 0]$ and $a_3 = [0, 1, 0, 1]$ . Find the coordinates $[\alpha_1, \alpha_2, \alpha_3]$ of $v$ in terms of $a_1, a_2, a_3$ , that is, the coefficients with $v = \alpha_1 a_1 + \alpha_2 a_2 + \alpha_3 a_3$ .