Solve {l}{y/8=x/10}{373x/1000=17+y}

$$\left\{ \begin{array} { l } { \frac { y } { 8 } = \frac { x } { 10 } } \\ { \frac { 373 x } { 1000 } = 17 + y } \end{array} \right.$$

Solve for y, x

$$x = -\frac{17000}{427} = -39\frac{347}{427} \approx -39.81264637$$

$$y = -\frac{13600}{427} = -31\frac{363}{427} \approx -31.850117096$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 8,10.

$$5y=4x$$

Divide both sides by 5.

$$y=\frac{1}{5}\times 4x$$

Multiply \frac{1}{5} times 4x.

$$y=\frac{4}{5}x$$

Substitute \frac{4x}{5} for y in the other equation, -1000y+373x=17000.

$$-1000\times \frac{4}{5}x+373x=17000$$

Multiply -1000 times \frac{4x}{5}.

$$-800x+373x=17000$$

Add -800x to 373x.

$$-427x=17000$$

Divide both sides by -427.

$$x=-\frac{17000}{427}$$

Substitute -\frac{17000}{427} for x in y=\frac{4}{5}x. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=\frac{4}{5}\left(-\frac{17000}{427}\right)$$

Multiply \frac{4}{5} times -\frac{17000}{427} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=-\frac{13600}{427}$$

The system is now solved.

$$y=-\frac{13600}{427},x=-\frac{17000}{427}$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 8,10.

$$5y=4x$$

Subtract 4x from both sides.

$$5y-4x=0$$

Consider the second equation. Multiply both sides of the equation by 1000.

$$373x=17000+1000y$$

Subtract 1000y from both sides.

$$373x-1000y=17000$$

Put the equations in standard form and then use matrices to solve the system of equations.

$$5y-4x=0,-1000y+373x=17000$$

Write the equations in matrix form.

$$\left(\begin{matrix}5&-4\\-1000&373\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\17000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}5&-4\\-1000&373\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-4\\-1000&373\end{matrix}\right))\left(\begin{matrix}5&-4\\-1000&373\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-1000&373\end{matrix}\right))\left(\begin{matrix}0\\17000\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-1000&373\end{matrix}\right))\left(\begin{matrix}0\\17000\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-1000&373\end{matrix}\right))\left(\begin{matrix}0\\17000\end{matrix}\right)$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{373}{5\times 373-\left(-4\left(-1000\right)\right)}&-\frac{-4}{5\times 373-\left(-4\left(-1000\right)\right)}\\-\frac{-1000}{5\times 373-\left(-4\left(-1000\right)\right)}&\frac{5}{5\times 373-\left(-4\left(-1000\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\17000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{373}{2135}&-\frac{4}{2135}\\-\frac{200}{427}&-\frac{1}{427}\end{matrix}\right)\left(\begin{matrix}0\\17000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{2135}\times 17000\\-\frac{1}{427}\times 17000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{13600}{427}\\-\frac{17000}{427}\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=-\frac{13600}{427},x=-\frac{17000}{427}$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 8,10.

$$5y=4x$$

Subtract 4x from both sides.

$$5y-4x=0$$

Consider the second equation. Multiply both sides of the equation by 1000.

$$373x=17000+1000y$$

Subtract 1000y from both sides.

$$373x-1000y=17000$$

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$5y-4x=0,-1000y+373x=17000$$

To make 5y and -1000y equal, multiply all terms on each side of the first equation by -1000 and all terms on each side of the second by 5.

$$-1000\times 5y-1000\left(-4\right)x=0,5\left(-1000\right)y+5\times 373x=5\times 17000$$

Simplify.

$$-5000y+4000x=0,-5000y+1865x=85000$$

Subtract -5000y+1865x=85000 from -5000y+4000x=0 by subtracting like terms on each side of the equal sign.

$$-5000y+5000y+4000x-1865x=-85000$$

Add -5000y to 5000y. Terms -5000y and 5000y cancel out, leaving an equation with only one variable that can be solved.

$$4000x-1865x=-85000$$

Add 4000x to -1865x.

$$2135x=-85000$$

Divide both sides by 2135.

$$x=-\frac{17000}{427}$$

Substitute -\frac{17000}{427} for x in -1000y+373x=17000. Because the resulting equation contains only one variable, you can solve for y directly.

$$-1000y+373\left(-\frac{17000}{427}\right)=17000$$

Multiply 373 times -\frac{17000}{427}.

$$-1000y-\frac{6341000}{427}=17000$$

Add \frac{6341000}{427} to both sides of the equation.

$$-1000y=\frac{13600000}{427}$$

Divide both sides by -1000.

$$y=-\frac{13600}{427}$$

The system is now solved.

$$y=-\frac{13600}{427},x=-\frac{17000}{427}$$