Solve {c}{902x-100y=2}{8*10^9-8*10^9x=15y-5x+4*10^3*y}

$$\left. \begin{array} { c } { 902 x - 100 y = 2 } \\ { 8 \times 10 ^ { 9 } - 8 \times 10 ^ { 9 } x = 15 y - 5 x + 4 \times 10 ^ { 3 } \times y } \end{array} \right.$$

Solve for x, y

$$x=\frac{80000000803}{80000362103}\approx 0.999995484$$

$$y = \frac{720000000001}{80000362103} = 8\frac{79997103177}{80000362103} \approx 8.999959264$$

Steps Using Substitution

Consider the second equation. Calculate 10 to the power of 9 and get 1000000000.

$$8\times 1000000000-8\times 10^{9}x=15y-5x+4\times 10^{3}y$$

Multiply 8 and 1000000000 to get 8000000000.

$$8000000000-8\times 10^{9}x=15y-5x+4\times 10^{3}y$$

Calculate 10 to the power of 9 and get 1000000000.

$$8000000000-8\times 1000000000x=15y-5x+4\times 10^{3}y$$

Multiply 8 and 1000000000 to get 8000000000.

$$8000000000-8000000000x=15y-5x+4\times 10^{3}y$$

Calculate 10 to the power of 3 and get 1000.

$$8000000000-8000000000x=15y-5x+4\times 1000y$$

Multiply 4 and 1000 to get 4000.

$$8000000000-8000000000x=15y-5x+4000y$$

Combine 15y and 4000y to get 4015y.

$$8000000000-8000000000x=4015y-5x$$

Subtract 4015y from both sides.

$$8000000000-8000000000x-4015y=-5x$$

Add 5x to both sides.

$$8000000000-8000000000x-4015y+5x=0$$

Combine -8000000000x and 5x to get -7999999995x.

$$8000000000-7999999995x-4015y=0$$

Subtract 8000000000 from both sides. Anything subtracted from zero gives its negation.

$$-7999999995x-4015y=-8000000000$$

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.

$$902x-100y=2,-7999999995x-4015y=-8000000000$$

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

$$902x-100y=2$$

Add 100y to both sides of the equation.

$$902x=100y+2$$

Divide both sides by 902.

$$x=\frac{1}{902}\left(100y+2\right)$$

Multiply \frac{1}{902} times 100y+2.

$$x=\frac{50}{451}y+\frac{1}{451}$$

Substitute \frac{50y+1}{451} for x in the other equation, -7999999995x-4015y=-8000000000.

$$-7999999995\left(\frac{50}{451}y+\frac{1}{451}\right)-4015y=-8000000000$$

Multiply -7999999995 times \frac{50y+1}{451}.

$$-\frac{399999999750}{451}y-\frac{7999999995}{451}-4015y=-8000000000$$

Add -\frac{399999999750y}{451} to -4015y.

$$-\frac{400001810515}{451}y-\frac{7999999995}{451}=-8000000000$$

Add \frac{7999999995}{451} to both sides of the equation.

$$-\frac{400001810515}{451}y=-\frac{3600000000005}{451}$$

Divide both sides of the equation by -\frac{400001810515}{451}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{720000000001}{80000362103}$$

Substitute \frac{720000000001}{80000362103} for y in x=\frac{50}{451}y+\frac{1}{451}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{50}{451}\times \frac{720000000001}{80000362103}+\frac{1}{451}$$

Multiply \frac{50}{451} times \frac{720000000001}{80000362103} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{36000000000050}{36080163308453}+\frac{1}{451}$$

Add \frac{1}{451} to \frac{36000000000050}{36080163308453} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{80000000803}{80000362103}$$

The system is now solved.

$$x=\frac{80000000803}{80000362103},y=\frac{720000000001}{80000362103}$$

Steps Using Matrices

Consider the second equation. Calculate 10 to the power of 9 and get 1000000000.

$$8\times 1000000000-8\times 10^{9}x=15y-5x+4\times 10^{3}y$$

Multiply 8 and 1000000000 to get 8000000000.

$$8000000000-8\times 10^{9}x=15y-5x+4\times 10^{3}y$$

Calculate 10 to the power of 9 and get 1000000000.

$$8000000000-8\times 1000000000x=15y-5x+4\times 10^{3}y$$

Multiply 8 and 1000000000 to get 8000000000.

$$8000000000-8000000000x=15y-5x+4\times 10^{3}y$$

Calculate 10 to the power of 3 and get 1000.

$$8000000000-8000000000x=15y-5x+4\times 1000y$$

Multiply 4 and 1000 to get 4000.

$$8000000000-8000000000x=15y-5x+4000y$$

Combine 15y and 4000y to get 4015y.

$$8000000000-8000000000x=4015y-5x$$

Subtract 4015y from both sides.

$$8000000000-8000000000x-4015y=-5x$$

Add 5x to both sides.

$$8000000000-8000000000x-4015y+5x=0$$

Combine -8000000000x and 5x to get -7999999995x.

$$8000000000-7999999995x-4015y=0$$

Subtract 8000000000 from both sides. Anything subtracted from zero gives its negation.

$$-7999999995x-4015y=-8000000000$$

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

$$902x-100y=2,-7999999995x-4015y=-8000000000$$

Write the equations in matrix form.

$$\left(\begin{matrix}902&-100\\-7999999995&-4015\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-8000000000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}902&-100\\-7999999995&-4015\end{matrix}\right).

$$inverse(\left(\begin{matrix}902&-100\\-7999999995&-4015\end{matrix}\right))\left(\begin{matrix}902&-100\\-7999999995&-4015\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}902&-100\\-7999999995&-4015\end{matrix}\right))\left(\begin{matrix}2\\-8000000000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}902&-100\\-7999999995&-4015\end{matrix}\right))\left(\begin{matrix}2\\-8000000000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}902&-100\\-7999999995&-4015\end{matrix}\right))\left(\begin{matrix}2\\-8000000000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4015}{902\left(-4015\right)-\left(-100\left(-7999999995\right)\right)}&-\frac{-100}{902\left(-4015\right)-\left(-100\left(-7999999995\right)\right)}\\-\frac{-7999999995}{902\left(-4015\right)-\left(-100\left(-7999999995\right)\right)}&\frac{902}{902\left(-4015\right)-\left(-100\left(-7999999995\right)\right)}\end{matrix}\right)\left(\begin{matrix}2\\-8000000000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{803}{160000724206}&-\frac{10}{80000362103}\\-\frac{1599999999}{160000724206}&-\frac{451}{400001810515}\end{matrix}\right)\left(\begin{matrix}2\\-8000000000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{803}{160000724206}\times 2-\frac{10}{80000362103}\left(-8000000000\right)\\-\frac{1599999999}{160000724206}\times 2-\frac{451}{400001810515}\left(-8000000000\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{80000000803}{80000362103}\\\frac{720000000001}{80000362103}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{80000000803}{80000362103},y=\frac{720000000001}{80000362103}$$

Steps Using Elimination

Consider the second equation. Calculate 10 to the power of 9 and get 1000000000.

$$8\times 1000000000-8\times 10^{9}x=15y-5x+4\times 10^{3}y$$

Multiply 8 and 1000000000 to get 8000000000.

$$8000000000-8\times 10^{9}x=15y-5x+4\times 10^{3}y$$

Calculate 10 to the power of 9 and get 1000000000.

$$8000000000-8\times 1000000000x=15y-5x+4\times 10^{3}y$$

Multiply 8 and 1000000000 to get 8000000000.

$$8000000000-8000000000x=15y-5x+4\times 10^{3}y$$

Calculate 10 to the power of 3 and get 1000.

$$8000000000-8000000000x=15y-5x+4\times 1000y$$

Multiply 4 and 1000 to get 4000.

$$8000000000-8000000000x=15y-5x+4000y$$

Combine 15y and 4000y to get 4015y.

$$8000000000-8000000000x=4015y-5x$$

Subtract 4015y from both sides.

$$8000000000-8000000000x-4015y=-5x$$

Add 5x to both sides.

$$8000000000-8000000000x-4015y+5x=0$$

Combine -8000000000x and 5x to get -7999999995x.

$$8000000000-7999999995x-4015y=0$$

Subtract 8000000000 from both sides. Anything subtracted from zero gives its negation.

$$-7999999995x-4015y=-8000000000$$

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.

$$902x-100y=2,-7999999995x-4015y=-8000000000$$

To make 902x and -7999999995x equal, multiply all terms on each side of the first equation by -7999999995 and all terms on each side of the second by 902.

$$-7999999995\times 902x-7999999995\left(-100\right)y=-7999999995\times 2,902\left(-7999999995\right)x+902\left(-4015\right)y=902\left(-8000000000\right)$$

Simplify.

$$-7215999995490x+799999999500y=-15999999990,-7215999995490x-3621530y=-7216000000000$$

Subtract -7215999995490x-3621530y=-7216000000000 from -7215999995490x+799999999500y=-15999999990 by subtracting like terms on each side of the equal sign.

$$-7215999995490x+7215999995490x+799999999500y+3621530y=-15999999990+7216000000000$$

Add -7215999995490x to 7215999995490x. Terms -7215999995490x and 7215999995490x cancel out, leaving an equation with only one variable that can be solved.

$$799999999500y+3621530y=-15999999990+7216000000000$$

Add 799999999500y to 3621530y.

$$800003621030y=-15999999990+7216000000000$$

Add -15999999990 to 7216000000000.

$$800003621030y=7200000000010$$

Divide both sides by 800003621030.

$$y=\frac{720000000001}{80000362103}$$

Substitute \frac{720000000001}{80000362103} for y in -7999999995x-4015y=-8000000000. Because the resulting equation contains only one variable, you can solve for x directly.

$$-7999999995x-4015\times \frac{720000000001}{80000362103}=-8000000000$$

Multiply -4015 times \frac{720000000001}{80000362103}.

$$-7999999995x-\frac{2890800000004015}{80000362103}=-8000000000$$

Add \frac{2890800000004015}{80000362103} to both sides of the equation.

$$-7999999995x=-\frac{640000006023999995985}{80000362103}$$

Divide both sides by -7999999995.

$$x=\frac{80000000803}{80000362103}$$

The system is now solved.

$$x=\frac{80000000803}{80000362103},y=\frac{720000000001}{80000362103}$$