Solve {l}{50-x/500=x/1000+x-y/2000}{x-y/2000=-frac{x}{750}+frac{y}{200}}

$$\left\{ \begin{array} { l } { \frac { 50 - x } { 500 } = \frac { x } { 1000 } + \frac { x - y } { 2000 } } \\ { \frac { x - y } { 2000 } = - \frac { x } { 750 } + \frac { y } { 200 } } \end{array} \right.$$

Solve for x, y

$$x=30$$

$$y=10$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 2000, the least common multiple of 500,1000,2000.

$$4\left(50-x\right)=2x+x-y$$

Use the distributive property to multiply 4 by 50-x.

$$200-4x=2x+x-y$$

Combine 2x and x to get 3x.

$$200-4x=3x-y$$

Subtract 3x from both sides.

$$200-4x-3x=-y$$

Combine -4x and -3x to get -7x.

$$200-7x=-y$$

Add y to both sides.

$$200-7x+y=0$$

Subtract 200 from both sides. Anything subtracted from zero gives its negation.

$$-7x+y=-200$$

Consider the second equation. Multiply both sides of the equation by 6000, the least common multiple of 2000,750,200.

$$3\left(x-y\right)=-8x+30y$$

Use the distributive property to multiply 3 by x-y.

$$3x-3y=-8x+30y$$

Add 8x to both sides.

$$3x-3y+8x=30y$$

Combine 3x and 8x to get 11x.

$$11x-3y=30y$$

Subtract 30y from both sides.

$$11x-3y-30y=0$$

Combine -3y and -30y to get -33y.

$$11x-33y=0$$

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$-7x+y=-200,11x-33y=0$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$-7x+y=-200$$

Subtract y from both sides of the equation.

$$-7x=-y-200$$

Divide both sides by -7.

$$x=-\frac{1}{7}\left(-y-200\right)$$

Multiply -\frac{1}{7} times -y-200.

$$x=\frac{1}{7}y+\frac{200}{7}$$

Substitute \frac{200+y}{7} for x in the other equation, 11x-33y=0.

$$11\left(\frac{1}{7}y+\frac{200}{7}\right)-33y=0$$

Multiply 11 times \frac{200+y}{7}.

$$\frac{11}{7}y+\frac{2200}{7}-33y=0$$

Add \frac{11y}{7} to -33y.

$$-\frac{220}{7}y+\frac{2200}{7}=0$$

Subtract \frac{2200}{7} from both sides of the equation.

$$-\frac{220}{7}y=-\frac{2200}{7}$$

Divide both sides of the equation by -\frac{220}{7}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=10$$

Substitute 10 for y in x=\frac{1}{7}y+\frac{200}{7}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{7}\times 10+\frac{200}{7}$$

Multiply \frac{1}{7} times 10.

$$x=\frac{10+200}{7}$$

Add \frac{200}{7} to \frac{10}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=30$$

The system is now solved.

$$x=30,y=10$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 2000, the least common multiple of 500,1000,2000.

$$4\left(50-x\right)=2x+x-y$$

Use the distributive property to multiply 4 by 50-x.

$$200-4x=2x+x-y$$

Combine 2x and x to get 3x.

$$200-4x=3x-y$$

Subtract 3x from both sides.

$$200-4x-3x=-y$$

Combine -4x and -3x to get -7x.

$$200-7x=-y$$

Add y to both sides.

$$200-7x+y=0$$

Subtract 200 from both sides. Anything subtracted from zero gives its negation.

$$-7x+y=-200$$

Consider the second equation. Multiply both sides of the equation by 6000, the least common multiple of 2000,750,200.

$$3\left(x-y\right)=-8x+30y$$

Use the distributive property to multiply 3 by x-y.

$$3x-3y=-8x+30y$$

Add 8x to both sides.

$$3x-3y+8x=30y$$

Combine 3x and 8x to get 11x.

$$11x-3y=30y$$

Subtract 30y from both sides.

$$11x-3y-30y=0$$

Combine -3y and -30y to get -33y.

$$11x-33y=0$$

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

$$-7x+y=-200,11x-33y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}-7&1\\11&-33\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-200\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-7&1\\11&-33\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7&1\\11&-33\end{matrix}\right))\left(\begin{matrix}-7&1\\11&-33\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&1\\11&-33\end{matrix}\right))\left(\begin{matrix}-200\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&1\\11&-33\end{matrix}\right))\left(\begin{matrix}-200\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&1\\11&-33\end{matrix}\right))\left(\begin{matrix}-200\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{33}{-7\left(-33\right)-11}&-\frac{1}{-7\left(-33\right)-11}\\-\frac{11}{-7\left(-33\right)-11}&-\frac{7}{-7\left(-33\right)-11}\end{matrix}\right)\left(\begin{matrix}-200\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{20}&-\frac{1}{220}\\-\frac{1}{20}&-\frac{7}{220}\end{matrix}\right)\left(\begin{matrix}-200\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{20}\left(-200\right)\\-\frac{1}{20}\left(-200\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\10\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=30,y=10$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 2000, the least common multiple of 500,1000,2000.

$$4\left(50-x\right)=2x+x-y$$

Use the distributive property to multiply 4 by 50-x.

$$200-4x=2x+x-y$$

Combine 2x and x to get 3x.

$$200-4x=3x-y$$

Subtract 3x from both sides.

$$200-4x-3x=-y$$

Combine -4x and -3x to get -7x.

$$200-7x=-y$$

Add y to both sides.

$$200-7x+y=0$$

Subtract 200 from both sides. Anything subtracted from zero gives its negation.

$$-7x+y=-200$$

Consider the second equation. Multiply both sides of the equation by 6000, the least common multiple of 2000,750,200.

$$3\left(x-y\right)=-8x+30y$$

Use the distributive property to multiply 3 by x-y.

$$3x-3y=-8x+30y$$

Add 8x to both sides.

$$3x-3y+8x=30y$$

Combine 3x and 8x to get 11x.

$$11x-3y=30y$$

Subtract 30y from both sides.

$$11x-3y-30y=0$$

Combine -3y and -30y to get -33y.

$$11x-33y=0$$

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.

$$-7x+y=-200,11x-33y=0$$

To make -7x and 11x equal, multiply all terms on each side of the first equation by 11 and all terms on each side of the second by -7.

$$11\left(-7\right)x+11y=11\left(-200\right),-7\times 11x-7\left(-33\right)y=0$$

Simplify.

$$-77x+11y=-2200,-77x+231y=0$$

Subtract -77x+231y=0 from -77x+11y=-2200 by subtracting like terms on each side of the equal sign.

$$-77x+77x+11y-231y=-2200$$

Add -77x to 77x. Terms -77x and 77x cancel out, leaving an equation with only one variable that can be solved.

$$11y-231y=-2200$$

Add 11y to -231y.

$$-220y=-2200$$

Divide both sides by -220.

$$y=10$$

Substitute 10 for y in 11x-33y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$11x-33\times 10=0$$

Multiply -33 times 10.

$$11x-330=0$$

Add 330 to both sides of the equation.

$$11x=330$$

Divide both sides by 11.

$$x=30$$

The system is now solved.

$$x=30,y=10$$