Solve {l}{5x-9y=139}{15x+2y=98}

$$\left. \begin{array} { l } { 5 x - 9 y = 139 } \\ { 15 x + 2 y = 98 } \end{array} \right.$$

Solve for x, y

$$x=8$$

$$y=-11$$

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.

$$5x-9y=139,15x+2y=98$$

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

$$5x-9y=139$$

Add 9y to both sides of the equation.

$$5x=9y+139$$

Divide both sides by 5.

$$x=\frac{1}{5}\left(9y+139\right)$$

Multiply \frac{1}{5} times 9y+139.

$$x=\frac{9}{5}y+\frac{139}{5}$$

Substitute \frac{9y+139}{5} for x in the other equation, 15x+2y=98.

$$15\left(\frac{9}{5}y+\frac{139}{5}\right)+2y=98$$

Multiply 15 times \frac{9y+139}{5}.

$$27y+417+2y=98$$

Add 27y to 2y.

$$29y+417=98$$

Subtract 417 from both sides of the equation.

$$29y=-319$$

Divide both sides by 29.

$$y=-11$$

Substitute -11 for y in x=\frac{9}{5}y+\frac{139}{5}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{9}{5}\left(-11\right)+\frac{139}{5}$$

Multiply \frac{9}{5} times -11.

$$x=\frac{-99+139}{5}$$

Add \frac{139}{5} to -\frac{99}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=8$$

The system is now solved.

$$x=8,y=-11$$

Steps Using Matrices

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

$$5x-9y=139,15x+2y=98$$

Write the equations in matrix form.

$$\left(\begin{matrix}5&-9\\15&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}139\\98\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}5&-9\\15&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-9\\15&2\end{matrix}\right))\left(\begin{matrix}5&-9\\15&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-9\\15&2\end{matrix}\right))\left(\begin{matrix}139\\98\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-9\\15&2\end{matrix}\right))\left(\begin{matrix}139\\98\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-9\\15&2\end{matrix}\right))\left(\begin{matrix}139\\98\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5\times 2-\left(-9\times 15\right)}&-\frac{-9}{5\times 2-\left(-9\times 15\right)}\\-\frac{15}{5\times 2-\left(-9\times 15\right)}&\frac{5}{5\times 2-\left(-9\times 15\right)}\end{matrix}\right)\left(\begin{matrix}139\\98\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{145}&\frac{9}{145}\\-\frac{3}{29}&\frac{1}{29}\end{matrix}\right)\left(\begin{matrix}139\\98\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{145}\times 139+\frac{9}{145}\times 98\\-\frac{3}{29}\times 139+\frac{1}{29}\times 98\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\-11\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=8,y=-11$$

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.

$$5x-9y=139,15x+2y=98$$

To make 5x and 15x equal, multiply all terms on each side of the first equation by 15 and all terms on each side of the second by 5.

$$15\times 5x+15\left(-9\right)y=15\times 139,5\times 15x+5\times 2y=5\times 98$$

Simplify.

$$75x-135y=2085,75x+10y=490$$

Subtract 75x+10y=490 from 75x-135y=2085 by subtracting like terms on each side of the equal sign.

$$75x-75x-135y-10y=2085-490$$

Add 75x to -75x. Terms 75x and -75x cancel out, leaving an equation with only one variable that can be solved.

$$-135y-10y=2085-490$$

Add -135y to -10y.

$$-145y=2085-490$$

Add 2085 to -490.

$$-145y=1595$$

Divide both sides by -145.

$$y=-11$$

Substitute -11 for y in 15x+2y=98. Because the resulting equation contains only one variable, you can solve for x directly.

$$15x+2\left(-11\right)=98$$

Multiply 2 times -11.

$$15x-22=98$$

Add 22 to both sides of the equation.

$$15x=120$$

Divide both sides by 15.

$$x=8$$

The system is now solved.

$$x=8,y=-11$$