Solve 97x+103y=497;103x+97y=503

$$97 x + 103 y = 497 ; 103 x + 97 y = 503$$

Solve for x, y

$$x=3$$

$$y=2$$

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.

$$97x+103y=497,103x+97y=503$$

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

$$97x+103y=497$$

Subtract 103y from both sides of the equation.

$$97x=-103y+497$$

Divide both sides by 97.

$$x=\frac{1}{97}\left(-103y+497\right)$$

Multiply \frac{1}{97} times -103y+497.

$$x=-\frac{103}{97}y+\frac{497}{97}$$

Substitute \frac{-103y+497}{97} for x in the other equation, 103x+97y=503.

$$103\left(-\frac{103}{97}y+\frac{497}{97}\right)+97y=503$$

Multiply 103 times \frac{-103y+497}{97}.

$$-\frac{10609}{97}y+\frac{51191}{97}+97y=503$$

Add -\frac{10609y}{97} to 97y.

$$-\frac{1200}{97}y+\frac{51191}{97}=503$$

Subtract \frac{51191}{97} from both sides of the equation.

$$-\frac{1200}{97}y=-\frac{2400}{97}$$

Divide both sides of the equation by -\frac{1200}{97}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

Substitute 2 for y in x=-\frac{103}{97}y+\frac{497}{97}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{103}{97}\times 2+\frac{497}{97}$$

Multiply -\frac{103}{97} times 2.

$$x=\frac{-206+497}{97}$$

Add \frac{497}{97} to -\frac{206}{97} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=3$$

The system is now solved.

$$x=3,y=2$$

Steps Using Matrices

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

$$97x+103y=497,103x+97y=503$$

Write the equations in matrix form.

$$\left(\begin{matrix}97&103\\103&97\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}497\\503\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}97&103\\103&97\end{matrix}\right).

$$inverse(\left(\begin{matrix}97&103\\103&97\end{matrix}\right))\left(\begin{matrix}97&103\\103&97\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}97&103\\103&97\end{matrix}\right))\left(\begin{matrix}497\\503\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}97&103\\103&97\end{matrix}\right))\left(\begin{matrix}497\\503\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}97&103\\103&97\end{matrix}\right))\left(\begin{matrix}497\\503\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{97}{97\times 97-103\times 103}&-\frac{103}{97\times 97-103\times 103}\\-\frac{103}{97\times 97-103\times 103}&\frac{97}{97\times 97-103\times 103}\end{matrix}\right)\left(\begin{matrix}497\\503\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{97}{1200}&\frac{103}{1200}\\\frac{103}{1200}&-\frac{97}{1200}\end{matrix}\right)\left(\begin{matrix}497\\503\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{97}{1200}\times 497+\frac{103}{1200}\times 503\\\frac{103}{1200}\times 497-\frac{97}{1200}\times 503\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\2\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=3,y=2$$

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.

$$97x+103y=497,103x+97y=503$$

To make 97x and 103x equal, multiply all terms on each side of the first equation by 103 and all terms on each side of the second by 97.

$$103\times 97x+103\times 103y=103\times 497,97\times 103x+97\times 97y=97\times 503$$

Simplify.

$$9991x+10609y=51191,9991x+9409y=48791$$

Subtract 9991x+9409y=48791 from 9991x+10609y=51191 by subtracting like terms on each side of the equal sign.

$$9991x-9991x+10609y-9409y=51191-48791$$

Add 9991x to -9991x. Terms 9991x and -9991x cancel out, leaving an equation with only one variable that can be solved.

$$10609y-9409y=51191-48791$$

Add 10609y to -9409y.

$$1200y=51191-48791$$

Add 51191 to -48791.

$$1200y=2400$$

Divide both sides by 1200.

$$y=2$$

Substitute 2 for y in 103x+97y=503. Because the resulting equation contains only one variable, you can solve for x directly.

$$103x+97\times 2=503$$

Multiply 97 times 2.

$$103x+194=503$$

Subtract 194 from both sides of the equation.

$$103x=309$$

Divide both sides by 103.

$$x=3$$

The system is now solved.

$$x=3,y=2$$