Solve {l}{1995x+1997y=5989}{1997x+1995y=5987}

$$\left\{ \begin{array} { l } { 1995 x + 1997 y = 5989 } \\ { 1997 x + 1995 y = 5987 } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=2$$

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.

$$1995x+1997y=5989,1997x+1995y=5987$$

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

$$1995x+1997y=5989$$

Subtract 1997y from both sides of the equation.

$$1995x=-1997y+5989$$

Divide both sides by 1995.

$$x=\frac{1}{1995}\left(-1997y+5989\right)$$

Multiply \frac{1}{1995} times -1997y+5989.

$$x=-\frac{1997}{1995}y+\frac{5989}{1995}$$

Substitute \frac{-1997y+5989}{1995} for x in the other equation, 1997x+1995y=5987.

$$1997\left(-\frac{1997}{1995}y+\frac{5989}{1995}\right)+1995y=5987$$

Multiply 1997 times \frac{-1997y+5989}{1995}.

$$-\frac{3988009}{1995}y+\frac{11960033}{1995}+1995y=5987$$

Add -\frac{3988009y}{1995} to 1995y.

$$-\frac{7984}{1995}y+\frac{11960033}{1995}=5987$$

Subtract \frac{11960033}{1995} from both sides of the equation.

$$-\frac{7984}{1995}y=-\frac{15968}{1995}$$

Divide both sides of the equation by -\frac{7984}{1995}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

Substitute 2 for y in x=-\frac{1997}{1995}y+\frac{5989}{1995}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1997}{1995}\times 2+\frac{5989}{1995}$$

Multiply -\frac{1997}{1995} times 2.

$$x=\frac{-3994+5989}{1995}$$

Add \frac{5989}{1995} to -\frac{3994}{1995} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=1$$

The system is now solved.

$$x=1,y=2$$

Steps Using Matrices

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

$$1995x+1997y=5989,1997x+1995y=5987$$

Write the equations in matrix form.

$$\left(\begin{matrix}1995&1997\\1997&1995\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5989\\5987\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1995&1997\\1997&1995\end{matrix}\right).

$$inverse(\left(\begin{matrix}1995&1997\\1997&1995\end{matrix}\right))\left(\begin{matrix}1995&1997\\1997&1995\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1995&1997\\1997&1995\end{matrix}\right))\left(\begin{matrix}5989\\5987\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1995&1997\\1997&1995\end{matrix}\right))\left(\begin{matrix}5989\\5987\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1995&1997\\1997&1995\end{matrix}\right))\left(\begin{matrix}5989\\5987\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1995}{1995\times 1995-1997\times 1997}&-\frac{1997}{1995\times 1995-1997\times 1997}\\-\frac{1997}{1995\times 1995-1997\times 1997}&\frac{1995}{1995\times 1995-1997\times 1997}\end{matrix}\right)\left(\begin{matrix}5989\\5987\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1995}{7984}&\frac{1997}{7984}\\\frac{1997}{7984}&-\frac{1995}{7984}\end{matrix}\right)\left(\begin{matrix}5989\\5987\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1995}{7984}\times 5989+\frac{1997}{7984}\times 5987\\\frac{1997}{7984}\times 5989-\frac{1995}{7984}\times 5987\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1,y=2$$

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.

$$1995x+1997y=5989,1997x+1995y=5987$$

To make 1995x and 1997x equal, multiply all terms on each side of the first equation by 1997 and all terms on each side of the second by 1995.

$$1997\times 1995x+1997\times 1997y=1997\times 5989,1995\times 1997x+1995\times 1995y=1995\times 5987$$

Simplify.

$$3984015x+3988009y=11960033,3984015x+3980025y=11944065$$

Subtract 3984015x+3980025y=11944065 from 3984015x+3988009y=11960033 by subtracting like terms on each side of the equal sign.

$$3984015x-3984015x+3988009y-3980025y=11960033-11944065$$

Add 3984015x to -3984015x. Terms 3984015x and -3984015x cancel out, leaving an equation with only one variable that can be solved.

$$3988009y-3980025y=11960033-11944065$$

Add 3988009y to -3980025y.

$$7984y=11960033-11944065$$

Add 11960033 to -11944065.

$$7984y=15968$$

Divide both sides by 7984.

$$y=2$$

Substitute 2 for y in 1997x+1995y=5987. Because the resulting equation contains only one variable, you can solve for x directly.

$$1997x+1995\times 2=5987$$

Multiply 1995 times 2.

$$1997x+3990=5987$$

Subtract 3990 from both sides of the equation.

$$1997x=1997$$

Divide both sides by 1997.

$$x=1$$

The system is now solved.

$$x=1,y=2$$