Solve {c}{d=ax+by+c/sqrt{a^2+b^2}}{a=3}{b=4(-1)}{c=18}{x=4}{y=5}

$$\left. \begin{array} { c } { d = \frac { a x + b y + c } { \sqrt { a ^ { 2 } + b ^ { 2 } } } } \\ { a = 3 } \\ { b = 4 ( - 1 ) } \\ { c = 18 } \\ { x = 4 } \\ { y = 5 } \end{array} \right.$$

Solve for d, a, x, b, y, c

$$x=4$$

$$y=5$$

$$d=2$$

$$a=3$$

$$b=-4$$

$$c=18$$

Solution Steps

Consider the third equation. Multiply 4 and -1 to get -4.

$$b=-4$$

Consider the first equation. Insert the known values of variables into the equation.

$$d=\frac{3\times 4-4\times 5+18}{\sqrt{3^{2}+\left(-4\right)^{2}}}$$

Do the multiplications.

$$d=\frac{12-20+18}{\sqrt{3^{2}+\left(-4\right)^{2}}}$$

Subtract 20 from 12 to get -8.

$$d=\frac{-8+18}{\sqrt{3^{2}+\left(-4\right)^{2}}}$$

Add -8 and 18 to get 10.

$$d=\frac{10}{\sqrt{3^{2}+\left(-4\right)^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$d=\frac{10}{\sqrt{9+\left(-4\right)^{2}}}$$

Calculate -4 to the power of 2 and get 16.

$$d=\frac{10}{\sqrt{9+16}}$$

Add 9 and 16 to get 25.

$$d=\frac{10}{\sqrt{25}}$$

Calculate the square root of 25 and get 5.

$$d=\frac{10}{5}$$

Divide 10 by 5 to get 2.

$$d=2$$

The system is now solved.

$$d=2 a=3 x=4 b=-4 y=5 c=18$$