Solve {l}{215x+170y=35}{185x+230y=365}

$$\left. \begin{array} { l } { 215 x + 170 y = 35 } \\ { 185 x + 230 y = 365 } \end{array} \right.$$

Solve for x, y

$$x=-3$$

$$y=4$$

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.

$$215x+170y=35,185x+230y=365$$

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

$$215x+170y=35$$

Subtract 170y from both sides of the equation.

$$215x=-170y+35$$

Divide both sides by 215.

$$x=\frac{1}{215}\left(-170y+35\right)$$

Multiply \frac{1}{215} times -170y+35.

$$x=-\frac{34}{43}y+\frac{7}{43}$$

Substitute \frac{-34y+7}{43} for x in the other equation, 185x+230y=365.

$$185\left(-\frac{34}{43}y+\frac{7}{43}\right)+230y=365$$

Multiply 185 times \frac{-34y+7}{43}.

$$-\frac{6290}{43}y+\frac{1295}{43}+230y=365$$

Add -\frac{6290y}{43} to 230y.

$$\frac{3600}{43}y+\frac{1295}{43}=365$$

Subtract \frac{1295}{43} from both sides of the equation.

$$\frac{3600}{43}y=\frac{14400}{43}$$

Divide both sides of the equation by \frac{3600}{43}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=4$$

Substitute 4 for y in x=-\frac{34}{43}y+\frac{7}{43}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{34}{43}\times 4+\frac{7}{43}$$

Multiply -\frac{34}{43} times 4.

$$x=\frac{-136+7}{43}$$

Add \frac{7}{43} to -\frac{136}{43} 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.

$$x=-3,y=4$$

Steps Using Matrices

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

$$215x+170y=35,185x+230y=365$$

Write the equations in matrix form.

$$\left(\begin{matrix}215&170\\185&230\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}35\\365\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}215&170\\185&230\end{matrix}\right).

$$inverse(\left(\begin{matrix}215&170\\185&230\end{matrix}\right))\left(\begin{matrix}215&170\\185&230\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}215&170\\185&230\end{matrix}\right))\left(\begin{matrix}35\\365\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}215&170\\185&230\end{matrix}\right))\left(\begin{matrix}35\\365\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}215&170\\185&230\end{matrix}\right))\left(\begin{matrix}35\\365\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{230}{215\times 230-170\times 185}&-\frac{170}{215\times 230-170\times 185}\\-\frac{185}{215\times 230-170\times 185}&\frac{215}{215\times 230-170\times 185}\end{matrix}\right)\left(\begin{matrix}35\\365\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{1800}&-\frac{17}{1800}\\-\frac{37}{3600}&\frac{43}{3600}\end{matrix}\right)\left(\begin{matrix}35\\365\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{1800}\times 35-\frac{17}{1800}\times 365\\-\frac{37}{3600}\times 35+\frac{43}{3600}\times 365\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\4\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-3,y=4$$

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.

$$215x+170y=35,185x+230y=365$$

To make 215x and 185x equal, multiply all terms on each side of the first equation by 185 and all terms on each side of the second by 215.

$$185\times 215x+185\times 170y=185\times 35,215\times 185x+215\times 230y=215\times 365$$

Simplify.

$$39775x+31450y=6475,39775x+49450y=78475$$

Subtract 39775x+49450y=78475 from 39775x+31450y=6475 by subtracting like terms on each side of the equal sign.

$$39775x-39775x+31450y-49450y=6475-78475$$

Add 39775x to -39775x. Terms 39775x and -39775x cancel out, leaving an equation with only one variable that can be solved.

$$31450y-49450y=6475-78475$$

Add 31450y to -49450y.

$$-18000y=6475-78475$$

Add 6475 to -78475.

$$-18000y=-72000$$

Divide both sides by -18000.

$$y=4$$

Substitute 4 for y in 185x+230y=365. Because the resulting equation contains only one variable, you can solve for x directly.

$$185x+230\times 4=365$$

Multiply 230 times 4.

$$185x+920=365$$

Subtract 920 from both sides of the equation.

$$185x=-555$$

Divide both sides by 185.

$$x=-3$$

The system is now solved.

$$x=-3,y=4$$