Solve {l}{2016x+2017y=20}{1008x+1008y=5}

$$\left\{ \begin{array} { l } { 2016 x + 2017 y = 20 } \\ { 1008 x + 1008 y = 5 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{10075}{1008} = -9\frac{1003}{1008} \approx -9.995039683$$

$$y=10$$

Steps Using Substitution

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.

$$2016x+2017y=20,1008x+1008y=5$$

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

$$2016x+2017y=20$$

Subtract 2017y from both sides of the equation.

$$2016x=-2017y+20$$

Divide both sides by 2016.

$$x=\frac{1}{2016}\left(-2017y+20\right)$$

Multiply \frac{1}{2016} times -2017y+20.

$$x=-\frac{2017}{2016}y+\frac{5}{504}$$

Substitute -\frac{2017y}{2016}+\frac{5}{504} for x in the other equation, 1008x+1008y=5.

$$1008\left(-\frac{2017}{2016}y+\frac{5}{504}\right)+1008y=5$$

Multiply 1008 times -\frac{2017y}{2016}+\frac{5}{504}.

$$-\frac{2017}{2}y+10+1008y=5$$

Add -\frac{2017y}{2} to 1008y.

$$-\frac{1}{2}y+10=5$$

Subtract 10 from both sides of the equation.

$$-\frac{1}{2}y=-5$$

Multiply both sides by -2.

$$y=10$$

Substitute 10 for y in x=-\frac{2017}{2016}y+\frac{5}{504}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{2017}{2016}\times 10+\frac{5}{504}$$

Multiply -\frac{2017}{2016} times 10.

$$x=-\frac{10085}{1008}+\frac{5}{504}$$

Add \frac{5}{504} to -\frac{10085}{1008} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{10075}{1008}$$

The system is now solved.

$$x=-\frac{10075}{1008},y=10$$

Steps Using Matrices

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

$$2016x+2017y=20,1008x+1008y=5$$

Write the equations in matrix form.

$$\left(\begin{matrix}2016&2017\\1008&1008\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\5\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2016&2017\\1008&1008\end{matrix}\right).

$$inverse(\left(\begin{matrix}2016&2017\\1008&1008\end{matrix}\right))\left(\begin{matrix}2016&2017\\1008&1008\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2016&2017\\1008&1008\end{matrix}\right))\left(\begin{matrix}20\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2016&2017\\1008&1008\end{matrix}\right))\left(\begin{matrix}20\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2016&2017\\1008&1008\end{matrix}\right))\left(\begin{matrix}20\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1008}{2016\times 1008-2017\times 1008}&-\frac{2017}{2016\times 1008-2017\times 1008}\\-\frac{1008}{2016\times 1008-2017\times 1008}&\frac{2016}{2016\times 1008-2017\times 1008}\end{matrix}\right)\left(\begin{matrix}20\\5\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&\frac{2017}{1008}\\1&-2\end{matrix}\right)\left(\begin{matrix}20\\5\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-20+\frac{2017}{1008}\times 5\\20-2\times 5\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10075}{1008}\\10\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{10075}{1008},y=10$$

Steps Using Elimination

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.

$$2016x+2017y=20,1008x+1008y=5$$

To make 2016x and 1008x equal, multiply all terms on each side of the first equation by 1008 and all terms on each side of the second by 2016.

$$1008\times 2016x+1008\times 2017y=1008\times 20,2016\times 1008x+2016\times 1008y=2016\times 5$$

Simplify.

$$2032128x+2033136y=20160,2032128x+2032128y=10080$$

Subtract 2032128x+2032128y=10080 from 2032128x+2033136y=20160 by subtracting like terms on each side of the equal sign.

$$2032128x-2032128x+2033136y-2032128y=20160-10080$$

Add 2032128x to -2032128x. Terms 2032128x and -2032128x cancel out, leaving an equation with only one variable that can be solved.

$$2033136y-2032128y=20160-10080$$

Add 2033136y to -2032128y.

$$1008y=20160-10080$$

Add 20160 to -10080.

$$1008y=10080$$

Divide both sides by 1008.

$$y=10$$

Substitute 10 for y in 1008x+1008y=5. Because the resulting equation contains only one variable, you can solve for x directly.

$$1008x+1008\times 10=5$$

Multiply 1008 times 10.

$$1008x+10080=5$$

Subtract 10080 from both sides of the equation.

$$1008x=-10075$$

Divide both sides by 1008.

$$x=-\frac{10075}{1008}$$

The system is now solved.

$$x=-\frac{10075}{1008},y=10$$