Solve {l}{99x+98y=97}{96x-95y=94}

$$\left\{ \begin{array} { l } { 99 x + 98 y = 97 } \\ { 96 x - 95 y = 94 } \end{array} \right.$$

Solve for x, y

$$x=\frac{18427}{18813}\approx 0.979482273$$

$$y=\frac{2}{6271}\approx 0.000318928$$

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.

$$99x+98y=97,96x-95y=94$$

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

$$99x+98y=97$$

Subtract 98y from both sides of the equation.

$$99x=-98y+97$$

Divide both sides by 99.

$$x=\frac{1}{99}\left(-98y+97\right)$$

Multiply \frac{1}{99} times -98y+97.

$$x=-\frac{98}{99}y+\frac{97}{99}$$

Substitute \frac{-98y+97}{99} for x in the other equation, 96x-95y=94.

$$96\left(-\frac{98}{99}y+\frac{97}{99}\right)-95y=94$$

Multiply 96 times \frac{-98y+97}{99}.

$$-\frac{3136}{33}y+\frac{3104}{33}-95y=94$$

Add -\frac{3136y}{33} to -95y.

$$-\frac{6271}{33}y+\frac{3104}{33}=94$$

Subtract \frac{3104}{33} from both sides of the equation.

$$-\frac{6271}{33}y=-\frac{2}{33}$$

Divide both sides of the equation by -\frac{6271}{33}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{2}{6271}$$

Substitute \frac{2}{6271} for y in x=-\frac{98}{99}y+\frac{97}{99}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{98}{99}\times \frac{2}{6271}+\frac{97}{99}$$

Multiply -\frac{98}{99} times \frac{2}{6271} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{196}{620829}+\frac{97}{99}$$

Add \frac{97}{99} to -\frac{196}{620829} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{18427}{18813}$$

The system is now solved.

$$x=\frac{18427}{18813},y=\frac{2}{6271}$$

Steps Using Matrices

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

$$99x+98y=97,96x-95y=94$$

Write the equations in matrix form.

$$\left(\begin{matrix}99&98\\96&-95\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}97\\94\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}99&98\\96&-95\end{matrix}\right).

$$inverse(\left(\begin{matrix}99&98\\96&-95\end{matrix}\right))\left(\begin{matrix}99&98\\96&-95\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}99&98\\96&-95\end{matrix}\right))\left(\begin{matrix}97\\94\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}99&98\\96&-95\end{matrix}\right))\left(\begin{matrix}97\\94\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}99&98\\96&-95\end{matrix}\right))\left(\begin{matrix}97\\94\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{95}{99\left(-95\right)-98\times 96}&-\frac{98}{99\left(-95\right)-98\times 96}\\-\frac{96}{99\left(-95\right)-98\times 96}&\frac{99}{99\left(-95\right)-98\times 96}\end{matrix}\right)\left(\begin{matrix}97\\94\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{95}{18813}&\frac{98}{18813}\\\frac{32}{6271}&-\frac{33}{6271}\end{matrix}\right)\left(\begin{matrix}97\\94\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{95}{18813}\times 97+\frac{98}{18813}\times 94\\\frac{32}{6271}\times 97-\frac{33}{6271}\times 94\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18427}{18813}\\\frac{2}{6271}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{18427}{18813},y=\frac{2}{6271}$$

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.

$$99x+98y=97,96x-95y=94$$

To make 99x and 96x equal, multiply all terms on each side of the first equation by 96 and all terms on each side of the second by 99.

$$96\times 99x+96\times 98y=96\times 97,99\times 96x+99\left(-95\right)y=99\times 94$$

Simplify.

$$9504x+9408y=9312,9504x-9405y=9306$$

Subtract 9504x-9405y=9306 from 9504x+9408y=9312 by subtracting like terms on each side of the equal sign.

$$9504x-9504x+9408y+9405y=9312-9306$$

Add 9504x to -9504x. Terms 9504x and -9504x cancel out, leaving an equation with only one variable that can be solved.

$$9408y+9405y=9312-9306$$

Add 9408y to 9405y.

$$18813y=9312-9306$$

Add 9312 to -9306.

$$18813y=6$$

Divide both sides by 18813.

$$y=\frac{2}{6271}$$

Substitute \frac{2}{6271} for y in 96x-95y=94. Because the resulting equation contains only one variable, you can solve for x directly.

$$96x-95\times \frac{2}{6271}=94$$

Multiply -95 times \frac{2}{6271}.

$$96x-\frac{190}{6271}=94$$

Add \frac{190}{6271} to both sides of the equation.

$$96x=\frac{589664}{6271}$$

Divide both sides by 96.

$$x=\frac{18427}{18813}$$

The system is now solved.

$$x=\frac{18427}{18813},y=\frac{2}{6271}$$