Solve {l}{0.0825x+0.7+1000(x+y)=2.11}{2000x-0.25+2000y+1000(x+y)=12}

$$\left. \begin{array} { l } { 0.0825 x + 0.7 + 1000 ( x + y ) = 2.11 } \\ { 2000 x - 0.25 + 2000 y + 1000 ( x + y ) = 12 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{3208}{99} = -32\frac{40}{99} \approx -32.404040404$$

$$y = \frac{12833617}{396000} = 32\frac{161617}{396000} \approx 32.408123737$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 1000 by x+y.

$$0.0825x+0.7+1000x+1000y=2.11$$

Combine 0.0825x and 1000x to get 1000.0825x.

$$1000.0825x+0.7+1000y=2.11$$

Subtract 0.7 from both sides.

$$1000.0825x+1000y=2.11-0.7$$

Subtract 0.7 from 2.11 to get 1.41.

$$1000.0825x+1000y=1.41$$

Consider the second equation. Use the distributive property to multiply 1000 by x+y.

$$2000x-0.25+2000y+1000x+1000y=12$$

Combine 2000x and 1000x to get 3000x.

$$3000x-0.25+2000y+1000y=12$$

Combine 2000y and 1000y to get 3000y.

$$3000x-0.25+3000y=12$$

Add 0.25 to both sides.

$$3000x+3000y=12+0.25$$

Add 12 and 0.25 to get 12.25.

$$3000x+3000y=12.25$$

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.

$$1000.0825x+1000y=1.41,3000x+3000y=12.25$$

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

$$1000.0825x+1000y=1.41$$

Subtract 1000y from both sides of the equation.

$$1000.0825x=-1000y+1.41$$

Divide both sides of the equation by 1000.0825, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{400}{400033}\left(-1000y+1.41\right)$$

Multiply \frac{400}{400033} times -1000y+1.41.

$$x=-\frac{400000}{400033}y+\frac{564}{400033}$$

Substitute \frac{-400000y+564}{400033} for x in the other equation, 3000x+3000y=12.25.

$$3000\left(-\frac{400000}{400033}y+\frac{564}{400033}\right)+3000y=12.25$$

Multiply 3000 times \frac{-400000y+564}{400033}.

$$-\frac{1200000000}{400033}y+\frac{1692000}{400033}+3000y=12.25$$

Add -\frac{1200000000y}{400033} to 3000y.

$$\frac{99000}{400033}y+\frac{1692000}{400033}=12.25$$

Subtract \frac{1692000}{400033} from both sides of the equation.

$$\frac{99000}{400033}y=\frac{12833617}{1600132}$$

Divide both sides of the equation by \frac{99000}{400033}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{12833617}{396000}$$

Substitute \frac{12833617}{396000} for y in x=-\frac{400000}{400033}y+\frac{564}{400033}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{400000}{400033}\times \frac{12833617}{396000}+\frac{564}{400033}$$

Multiply -\frac{400000}{400033} times \frac{12833617}{396000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{1283361700}{39603267}+\frac{564}{400033}$$

Add \frac{564}{400033} to -\frac{1283361700}{39603267} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{3208}{99}$$

The system is now solved.

$$x=-\frac{3208}{99},y=\frac{12833617}{396000}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 1000 by x+y.

$$0.0825x+0.7+1000x+1000y=2.11$$

Combine 0.0825x and 1000x to get 1000.0825x.

$$1000.0825x+0.7+1000y=2.11$$

Subtract 0.7 from both sides.

$$1000.0825x+1000y=2.11-0.7$$

Subtract 0.7 from 2.11 to get 1.41.

$$1000.0825x+1000y=1.41$$

Consider the second equation. Use the distributive property to multiply 1000 by x+y.

$$2000x-0.25+2000y+1000x+1000y=12$$

Combine 2000x and 1000x to get 3000x.

$$3000x-0.25+2000y+1000y=12$$

Combine 2000y and 1000y to get 3000y.

$$3000x-0.25+3000y=12$$

Add 0.25 to both sides.

$$3000x+3000y=12+0.25$$

Add 12 and 0.25 to get 12.25.

$$3000x+3000y=12.25$$

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

$$1000.0825x+1000y=1.41,3000x+3000y=12.25$$

Write the equations in matrix form.

$$\left(\begin{matrix}1000.0825&1000\\3000&3000\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1.41\\12.25\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1000.0825&1000\\3000&3000\end{matrix}\right).

$$inverse(\left(\begin{matrix}1000.0825&1000\\3000&3000\end{matrix}\right))\left(\begin{matrix}1000.0825&1000\\3000&3000\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1000.0825&1000\\3000&3000\end{matrix}\right))\left(\begin{matrix}1.41\\12.25\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1000.0825&1000\\3000&3000\end{matrix}\right))\left(\begin{matrix}1.41\\12.25\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1000.0825&1000\\3000&3000\end{matrix}\right))\left(\begin{matrix}1.41\\12.25\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3000}{1000.0825\times 3000-1000\times 3000}&-\frac{1000}{1000.0825\times 3000-1000\times 3000}\\-\frac{3000}{1000.0825\times 3000-1000\times 3000}&\frac{1000.0825}{1000.0825\times 3000-1000\times 3000}\end{matrix}\right)\left(\begin{matrix}1.41\\12.25\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{400}{33}&-\frac{400}{99}\\-\frac{400}{33}&\frac{400033}{99000}\end{matrix}\right)\left(\begin{matrix}1.41\\12.25\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{400}{33}\times 1.41-\frac{400}{99}\times 12.25\\-\frac{400}{33}\times 1.41+\frac{400033}{99000}\times 12.25\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3208}{99}\\\frac{12833617}{396000}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{3208}{99},y=\frac{12833617}{396000}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 1000 by x+y.

$$0.0825x+0.7+1000x+1000y=2.11$$

Combine 0.0825x and 1000x to get 1000.0825x.

$$1000.0825x+0.7+1000y=2.11$$

Subtract 0.7 from both sides.

$$1000.0825x+1000y=2.11-0.7$$

Subtract 0.7 from 2.11 to get 1.41.

$$1000.0825x+1000y=1.41$$

Consider the second equation. Use the distributive property to multiply 1000 by x+y.

$$2000x-0.25+2000y+1000x+1000y=12$$

Combine 2000x and 1000x to get 3000x.

$$3000x-0.25+2000y+1000y=12$$

Combine 2000y and 1000y to get 3000y.

$$3000x-0.25+3000y=12$$

Add 0.25 to both sides.

$$3000x+3000y=12+0.25$$

Add 12 and 0.25 to get 12.25.

$$3000x+3000y=12.25$$

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.

$$1000.0825x+1000y=1.41,3000x+3000y=12.25$$

To make \frac{400033x}{400} and 3000x equal, multiply all terms on each side of the first equation by 3000 and all terms on each side of the second by 1000.0825.

$$3000\times 1000.0825x+3000\times 1000y=3000\times 1.41,1000.0825\times 3000x+1000.0825\times 3000y=1000.0825\times 12.25$$

Simplify.

$$3000247.5x+3000000y=4230,3000247.5x+3000247.5y=12251.010625$$

Subtract 3000247.5x+3000247.5y=12251.010625 from 3000247.5x+3000000y=4230 by subtracting like terms on each side of the equal sign.

$$3000247.5x-3000247.5x+3000000y-3000247.5y=4230-12251.010625$$

Add \frac{6000495x}{2} to -\frac{6000495x}{2}. Terms \frac{6000495x}{2} and -\frac{6000495x}{2} cancel out, leaving an equation with only one variable that can be solved.

$$3000000y-3000247.5y=4230-12251.010625$$

Add 3000000y to -\frac{6000495y}{2}.

$$-247.5y=4230-12251.010625$$

Add 4230 to -12251.010625.

$$-247.5y=-8021.010625$$

Divide both sides of the equation by -247.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{12833617}{396000}$$

Substitute \frac{12833617}{396000} for y in 3000x+3000y=12.25. Because the resulting equation contains only one variable, you can solve for x directly.

$$3000x+3000\times \frac{12833617}{396000}=12.25$$

Multiply 3000 times \frac{12833617}{396000}.

$$3000x+\frac{12833617}{132}=12.25$$

Subtract \frac{12833617}{132} from both sides of the equation.

$$3000x=-\frac{3208000}{33}$$

Divide both sides by 3000.

$$x=-\frac{3208}{99}$$

The system is now solved.

$$x=-\frac{3208}{99},y=\frac{12833617}{396000}$$