Solve {l}{15x+19y=-31}{15(104-12x/17)+19y=-31}

$$\left. \begin{array} { l } { 15 x + 19 y = - 31 } \\ { 15 ( \frac { 104 - 12 x } { 17 } ) + 19 y = - 31 } \end{array} \right.$$

Solve for x, y

$$x = \frac{104}{29} = 3\frac{17}{29} \approx 3.586206897$$

$$y = -\frac{2459}{551} = -4\frac{255}{551} \approx -4.462794918$$

Steps Using Substitution

Consider the second equation. Multiply both sides of the equation by 17.

$$15\left(104-12x\right)+323y=-527$$

Use the distributive property to multiply 15 by 104-12x.

$$1560-180x+323y=-527$$

Subtract 1560 from both sides.

$$-180x+323y=-527-1560$$

Subtract 1560 from -527 to get -2087.

$$-180x+323y=-2087$$

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.

$$15x+19y=-31,-180x+323y=-2087$$

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

$$15x+19y=-31$$

Subtract 19y from both sides of the equation.

$$15x=-19y-31$$

Divide both sides by 15.

$$x=\frac{1}{15}\left(-19y-31\right)$$

Multiply \frac{1}{15} times -19y-31.

$$x=-\frac{19}{15}y-\frac{31}{15}$$

Substitute \frac{-19y-31}{15} for x in the other equation, -180x+323y=-2087.

$$-180\left(-\frac{19}{15}y-\frac{31}{15}\right)+323y=-2087$$

Multiply -180 times \frac{-19y-31}{15}.

$$228y+372+323y=-2087$$

Add 228y to 323y.

$$551y+372=-2087$$

Subtract 372 from both sides of the equation.

$$551y=-2459$$

Divide both sides by 551.

$$y=-\frac{2459}{551}$$

Substitute -\frac{2459}{551} for y in x=-\frac{19}{15}y-\frac{31}{15}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{19}{15}\left(-\frac{2459}{551}\right)-\frac{31}{15}$$

Multiply -\frac{19}{15} times -\frac{2459}{551} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{2459}{435}-\frac{31}{15}$$

Add -\frac{31}{15} to \frac{2459}{435} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{104}{29}$$

The system is now solved.

$$x=\frac{104}{29},y=-\frac{2459}{551}$$

Steps Using Matrices

Consider the second equation. Multiply both sides of the equation by 17.

$$15\left(104-12x\right)+323y=-527$$

Use the distributive property to multiply 15 by 104-12x.

$$1560-180x+323y=-527$$

Subtract 1560 from both sides.

$$-180x+323y=-527-1560$$

Subtract 1560 from -527 to get -2087.

$$-180x+323y=-2087$$

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

$$15x+19y=-31,-180x+323y=-2087$$

Write the equations in matrix form.

$$\left(\begin{matrix}15&19\\-180&323\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-31\\-2087\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}15&19\\-180&323\end{matrix}\right).

$$inverse(\left(\begin{matrix}15&19\\-180&323\end{matrix}\right))\left(\begin{matrix}15&19\\-180&323\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&19\\-180&323\end{matrix}\right))\left(\begin{matrix}-31\\-2087\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&19\\-180&323\end{matrix}\right))\left(\begin{matrix}-31\\-2087\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&19\\-180&323\end{matrix}\right))\left(\begin{matrix}-31\\-2087\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{323}{15\times 323-19\left(-180\right)}&-\frac{19}{15\times 323-19\left(-180\right)}\\-\frac{-180}{15\times 323-19\left(-180\right)}&\frac{15}{15\times 323-19\left(-180\right)}\end{matrix}\right)\left(\begin{matrix}-31\\-2087\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{435}&-\frac{1}{435}\\\frac{12}{551}&\frac{1}{551}\end{matrix}\right)\left(\begin{matrix}-31\\-2087\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{435}\left(-31\right)-\frac{1}{435}\left(-2087\right)\\\frac{12}{551}\left(-31\right)+\frac{1}{551}\left(-2087\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{104}{29}\\-\frac{2459}{551}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{104}{29},y=-\frac{2459}{551}$$

Steps Using Elimination

Consider the second equation. Multiply both sides of the equation by 17.

$$15\left(104-12x\right)+323y=-527$$

Use the distributive property to multiply 15 by 104-12x.

$$1560-180x+323y=-527$$

Subtract 1560 from both sides.

$$-180x+323y=-527-1560$$

Subtract 1560 from -527 to get -2087.

$$-180x+323y=-2087$$

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.

$$15x+19y=-31,-180x+323y=-2087$$

To make 15x and -180x equal, multiply all terms on each side of the first equation by -180 and all terms on each side of the second by 15.

$$-180\times 15x-180\times 19y=-180\left(-31\right),15\left(-180\right)x+15\times 323y=15\left(-2087\right)$$

Simplify.

$$-2700x-3420y=5580,-2700x+4845y=-31305$$

Subtract -2700x+4845y=-31305 from -2700x-3420y=5580 by subtracting like terms on each side of the equal sign.

$$-2700x+2700x-3420y-4845y=5580+31305$$

Add -2700x to 2700x. Terms -2700x and 2700x cancel out, leaving an equation with only one variable that can be solved.

$$-3420y-4845y=5580+31305$$

Add -3420y to -4845y.

$$-8265y=5580+31305$$

Add 5580 to 31305.

$$-8265y=36885$$

Divide both sides by -8265.

$$y=-\frac{2459}{551}$$

Substitute -\frac{2459}{551} for y in -180x+323y=-2087. Because the resulting equation contains only one variable, you can solve for x directly.

$$-180x+323\left(-\frac{2459}{551}\right)=-2087$$

Multiply 323 times -\frac{2459}{551}.

$$-180x-\frac{41803}{29}=-2087$$

Add \frac{41803}{29} to both sides of the equation.

$$-180x=-\frac{18720}{29}$$

Divide both sides by -180.

$$x=\frac{104}{29}$$

The system is now solved.

$$x=\frac{104}{29},y=-\frac{2459}{551}$$