Solve {l}{220x+108+100y=352}{600y+220x+108=316}

$$\left\{ \begin{array} { l } { 220 x + 108 + 100 y = 352 } \\ { 600 y + 220 x + 108 = 316 } \end{array} \right.$$

Solve for x, y

$$x = \frac{314}{275} = 1\frac{39}{275} \approx 1.141818182$$

$$y=-\frac{9}{125}=-0.072$$

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.

$$220x+100y+108=352,220x+600y+108=316$$

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

$$220x+100y+108=352$$

Subtract 108 from both sides of the equation.

$$220x+100y=244$$

Subtract 100y from both sides of the equation.

$$220x=-100y+244$$

Divide both sides by 220.

$$x=\frac{1}{220}\left(-100y+244\right)$$

Multiply \frac{1}{220} times -100y+244.

$$x=-\frac{5}{11}y+\frac{61}{55}$$

Substitute -\frac{5y}{11}+\frac{61}{55} for x in the other equation, 220x+600y+108=316.

$$220\left(-\frac{5}{11}y+\frac{61}{55}\right)+600y+108=316$$

Multiply 220 times -\frac{5y}{11}+\frac{61}{55}.

$$-100y+244+600y+108=316$$

Add -100y to 600y.

$$500y+244+108=316$$

Add 244 to 108.

$$500y+352=316$$

Subtract 352 from both sides of the equation.

$$500y=-36$$

Divide both sides by 500.

$$y=-\frac{9}{125}$$

Substitute -\frac{9}{125} for y in x=-\frac{5}{11}y+\frac{61}{55}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{5}{11}\left(-\frac{9}{125}\right)+\frac{61}{55}$$

Multiply -\frac{5}{11} times -\frac{9}{125} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{9}{275}+\frac{61}{55}$$

Add \frac{61}{55} to \frac{9}{275} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{314}{275}$$

The system is now solved.

$$x=\frac{314}{275},y=-\frac{9}{125}$$

Steps Using Matrices

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

$$220x+100y+108=352,220x+600y+108=316$$

Write the equations in matrix form.

$$\left(\begin{matrix}220&100\\220&600\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}244\\208\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}220&100\\220&600\end{matrix}\right).

$$inverse(\left(\begin{matrix}220&100\\220&600\end{matrix}\right))\left(\begin{matrix}220&100\\220&600\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}220&100\\220&600\end{matrix}\right))\left(\begin{matrix}244\\208\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}220&100\\220&600\end{matrix}\right))\left(\begin{matrix}244\\208\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}220&100\\220&600\end{matrix}\right))\left(\begin{matrix}244\\208\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{600}{220\times 600-100\times 220}&-\frac{100}{220\times 600-100\times 220}\\-\frac{220}{220\times 600-100\times 220}&\frac{220}{220\times 600-100\times 220}\end{matrix}\right)\left(\begin{matrix}244\\208\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{550}&-\frac{1}{1100}\\-\frac{1}{500}&\frac{1}{500}\end{matrix}\right)\left(\begin{matrix}244\\208\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{550}\times 244-\frac{1}{1100}\times 208\\-\frac{1}{500}\times 244+\frac{1}{500}\times 208\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{314}{275}\\-\frac{9}{125}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{314}{275},y=-\frac{9}{125}$$

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.

$$220x+100y+108=352,220x+600y+108=316$$

Subtract 220x+600y+108=316 from 220x+100y+108=352 by subtracting like terms on each side of the equal sign.

$$220x-220x+100y-600y+108-108=352-316$$

Add 220x to -220x. Terms 220x and -220x cancel out, leaving an equation with only one variable that can be solved.

$$100y-600y+108-108=352-316$$

Add 100y to -600y.

$$-500y+108-108=352-316$$

Add 108 to -108.

$$-500y=352-316$$

Add 352 to -316.

$$-500y=36$$

Divide both sides by -500.

$$y=-\frac{9}{125}$$

Substitute -\frac{9}{125} for y in 220x+600y+108=316. Because the resulting equation contains only one variable, you can solve for x directly.

$$220x+600\left(-\frac{9}{125}\right)+108=316$$

Multiply 600 times -\frac{9}{125}.

$$220x-\frac{216}{5}+108=316$$

Add -\frac{216}{5} to 108.

$$220x+\frac{324}{5}=316$$

Subtract \frac{324}{5} from both sides of the equation.

$$220x=\frac{1256}{5}$$

Divide both sides by 220.

$$x=\frac{314}{275}$$

The system is now solved.

$$x=\frac{314}{275},y=-\frac{9}{125}$$