Solve {l}{4x+9y=900*(900-x-y)}{5x+11y=1045-(1045-x-y)}

$$\left\{ \begin{array} { l } { 4 x + 9 y = 900 \cdot ( 900 - x - y ) } \\ { 5 x + 11 y = 1045 - ( 1045 - x - y ) } \end{array} \right.$$

Solve for x, y

$$x = \frac{2025000}{1351} = 1498\frac{1202}{1351} \approx 1498.889711325$$

$$y = -\frac{810000}{1351} = -599\frac{751}{1351} \approx -599.55588453$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 900 by 900-x-y.

$$4x+9y=810000-900x-900y$$

Add 900x to both sides.

$$4x+9y+900x=810000-900y$$

Combine 4x and 900x to get 904x.

$$904x+9y=810000-900y$$

Add 900y to both sides.

$$904x+9y+900y=810000$$

Combine 9y and 900y to get 909y.

$$904x+909y=810000$$

Consider the second equation. To find the opposite of 1045-x-y, find the opposite of each term.

$$5x+11y=1045-1045+x+y$$

Subtract 1045 from 1045 to get 0.

$$5x+11y=x+y$$

Subtract x from both sides.

$$5x+11y-x=y$$

Combine 5x and -x to get 4x.

$$4x+11y=y$$

Subtract y from both sides.

$$4x+11y-y=0$$

Combine 11y and -y to get 10y.

$$4x+10y=0$$

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.

$$904x+909y=810000,4x+10y=0$$

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

$$904x+909y=810000$$

Subtract 909y from both sides of the equation.

$$904x=-909y+810000$$

Divide both sides by 904.

$$x=\frac{1}{904}\left(-909y+810000\right)$$

Multiply \frac{1}{904} times -909y+810000.

$$x=-\frac{909}{904}y+\frac{101250}{113}$$

Substitute -\frac{909y}{904}+\frac{101250}{113} for x in the other equation, 4x+10y=0.

$$4\left(-\frac{909}{904}y+\frac{101250}{113}\right)+10y=0$$

Multiply 4 times -\frac{909y}{904}+\frac{101250}{113}.

$$-\frac{909}{226}y+\frac{405000}{113}+10y=0$$

Add -\frac{909y}{226} to 10y.

$$\frac{1351}{226}y+\frac{405000}{113}=0$$

Subtract \frac{405000}{113} from both sides of the equation.

$$\frac{1351}{226}y=-\frac{405000}{113}$$

Divide both sides of the equation by \frac{1351}{226}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{810000}{1351}$$

Substitute -\frac{810000}{1351} for y in x=-\frac{909}{904}y+\frac{101250}{113}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{909}{904}\left(-\frac{810000}{1351}\right)+\frac{101250}{113}$$

Multiply -\frac{909}{904} times -\frac{810000}{1351} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{92036250}{152663}+\frac{101250}{113}$$

Add \frac{101250}{113} to \frac{92036250}{152663} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{2025000}{1351}$$

The system is now solved.

$$x=\frac{2025000}{1351},y=-\frac{810000}{1351}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 900 by 900-x-y.

$$4x+9y=810000-900x-900y$$

Add 900x to both sides.

$$4x+9y+900x=810000-900y$$

Combine 4x and 900x to get 904x.

$$904x+9y=810000-900y$$

Add 900y to both sides.

$$904x+9y+900y=810000$$

Combine 9y and 900y to get 909y.

$$904x+909y=810000$$

Consider the second equation. To find the opposite of 1045-x-y, find the opposite of each term.

$$5x+11y=1045-1045+x+y$$

Subtract 1045 from 1045 to get 0.

$$5x+11y=x+y$$

Subtract x from both sides.

$$5x+11y-x=y$$

Combine 5x and -x to get 4x.

$$4x+11y=y$$

Subtract y from both sides.

$$4x+11y-y=0$$

Combine 11y and -y to get 10y.

$$4x+10y=0$$

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

$$904x+909y=810000,4x+10y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}904&909\\4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}810000\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}904&909\\4&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}904&909\\4&10\end{matrix}\right))\left(\begin{matrix}904&909\\4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}904&909\\4&10\end{matrix}\right))\left(\begin{matrix}810000\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}904&909\\4&10\end{matrix}\right))\left(\begin{matrix}810000\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}904&909\\4&10\end{matrix}\right))\left(\begin{matrix}810000\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{904\times 10-909\times 4}&-\frac{909}{904\times 10-909\times 4}\\-\frac{4}{904\times 10-909\times 4}&\frac{904}{904\times 10-909\times 4}\end{matrix}\right)\left(\begin{matrix}810000\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2702}&-\frac{909}{5404}\\-\frac{1}{1351}&\frac{226}{1351}\end{matrix}\right)\left(\begin{matrix}810000\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2702}\times 810000\\-\frac{1}{1351}\times 810000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2025000}{1351}\\-\frac{810000}{1351}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{2025000}{1351},y=-\frac{810000}{1351}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 900 by 900-x-y.

$$4x+9y=810000-900x-900y$$

Add 900x to both sides.

$$4x+9y+900x=810000-900y$$

Combine 4x and 900x to get 904x.

$$904x+9y=810000-900y$$

Add 900y to both sides.

$$904x+9y+900y=810000$$

Combine 9y and 900y to get 909y.

$$904x+909y=810000$$

Consider the second equation. To find the opposite of 1045-x-y, find the opposite of each term.

$$5x+11y=1045-1045+x+y$$

Subtract 1045 from 1045 to get 0.

$$5x+11y=x+y$$

Subtract x from both sides.

$$5x+11y-x=y$$

Combine 5x and -x to get 4x.

$$4x+11y=y$$

Subtract y from both sides.

$$4x+11y-y=0$$

Combine 11y and -y to get 10y.

$$4x+10y=0$$

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.

$$904x+909y=810000,4x+10y=0$$

To make 904x and 4x equal, multiply all terms on each side of the first equation by 4 and all terms on each side of the second by 904.

$$4\times 904x+4\times 909y=4\times 810000,904\times 4x+904\times 10y=0$$

Simplify.

$$3616x+3636y=3240000,3616x+9040y=0$$

Subtract 3616x+9040y=0 from 3616x+3636y=3240000 by subtracting like terms on each side of the equal sign.

$$3616x-3616x+3636y-9040y=3240000$$

Add 3616x to -3616x. Terms 3616x and -3616x cancel out, leaving an equation with only one variable that can be solved.

$$3636y-9040y=3240000$$

Add 3636y to -9040y.

$$-5404y=3240000$$

Divide both sides by -5404.

$$y=-\frac{810000}{1351}$$

Substitute -\frac{810000}{1351} for y in 4x+10y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$4x+10\left(-\frac{810000}{1351}\right)=0$$

Multiply 10 times -\frac{810000}{1351}.

$$4x-\frac{8100000}{1351}=0$$

Add \frac{8100000}{1351} to both sides of the equation.

$$4x=\frac{8100000}{1351}$$

Divide both sides by 4.

$$x=\frac{2025000}{1351}$$

The system is now solved.

$$x=\frac{2025000}{1351},y=-\frac{810000}{1351}$$