Solve {l}{37y+43x=23}{43y+37x=27}

$$\left. \begin{array} { l } { 37 y + 43 x = 23 } \\ { 43 y + 37 x = 27 } \end{array} \right.$$

Solve for y, x

$$x=-\frac{1}{48}\approx -0.020833333$$

$$y=\frac{31}{48}\approx 0.645833333$$

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.

$$37y+43x=23,43y+37x=27$$

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

$$37y+43x=23$$

Subtract 43x from both sides of the equation.

$$37y=-43x+23$$

Divide both sides by 37.

$$y=\frac{1}{37}\left(-43x+23\right)$$

Multiply \frac{1}{37} times -43x+23.

$$y=-\frac{43}{37}x+\frac{23}{37}$$

Substitute \frac{-43x+23}{37} for y in the other equation, 43y+37x=27.

$$43\left(-\frac{43}{37}x+\frac{23}{37}\right)+37x=27$$

Multiply 43 times \frac{-43x+23}{37}.

$$-\frac{1849}{37}x+\frac{989}{37}+37x=27$$

Add -\frac{1849x}{37} to 37x.

$$-\frac{480}{37}x+\frac{989}{37}=27$$

Subtract \frac{989}{37} from both sides of the equation.

$$-\frac{480}{37}x=\frac{10}{37}$$

Divide both sides of the equation by -\frac{480}{37}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{1}{48}$$

Substitute -\frac{1}{48} for x in y=-\frac{43}{37}x+\frac{23}{37}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=-\frac{43}{37}\left(-\frac{1}{48}\right)+\frac{23}{37}$$

Multiply -\frac{43}{37} times -\frac{1}{48} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{43}{1776}+\frac{23}{37}$$

Add \frac{23}{37} to \frac{43}{1776} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{31}{48}$$

The system is now solved.

$$y=\frac{31}{48},x=-\frac{1}{48}$$

Steps Using Matrices

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

$$37y+43x=23,43y+37x=27$$

Write the equations in matrix form.

$$\left(\begin{matrix}37&43\\43&37\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}23\\27\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}37&43\\43&37\end{matrix}\right).

$$inverse(\left(\begin{matrix}37&43\\43&37\end{matrix}\right))\left(\begin{matrix}37&43\\43&37\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}37&43\\43&37\end{matrix}\right))\left(\begin{matrix}23\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}37&43\\43&37\end{matrix}\right))\left(\begin{matrix}23\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}37&43\\43&37\end{matrix}\right))\left(\begin{matrix}23\\27\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{37}{37\times 37-43\times 43}&-\frac{43}{37\times 37-43\times 43}\\-\frac{43}{37\times 37-43\times 43}&\frac{37}{37\times 37-43\times 43}\end{matrix}\right)\left(\begin{matrix}23\\27\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{37}{480}&\frac{43}{480}\\\frac{43}{480}&-\frac{37}{480}\end{matrix}\right)\left(\begin{matrix}23\\27\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{37}{480}\times 23+\frac{43}{480}\times 27\\\frac{43}{480}\times 23-\frac{37}{480}\times 27\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{31}{48}\\-\frac{1}{48}\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=\frac{31}{48},x=-\frac{1}{48}$$

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.

$$37y+43x=23,43y+37x=27$$

To make 37y and 43y equal, multiply all terms on each side of the first equation by 43 and all terms on each side of the second by 37.

$$43\times 37y+43\times 43x=43\times 23,37\times 43y+37\times 37x=37\times 27$$

Simplify.

$$1591y+1849x=989,1591y+1369x=999$$

Subtract 1591y+1369x=999 from 1591y+1849x=989 by subtracting like terms on each side of the equal sign.

$$1591y-1591y+1849x-1369x=989-999$$

Add 1591y to -1591y. Terms 1591y and -1591y cancel out, leaving an equation with only one variable that can be solved.

$$1849x-1369x=989-999$$

Add 1849x to -1369x.

$$480x=989-999$$

Add 989 to -999.

$$480x=-10$$

Divide both sides by 480.

$$x=-\frac{1}{48}$$

Substitute -\frac{1}{48} for x in 43y+37x=27. Because the resulting equation contains only one variable, you can solve for y directly.

$$43y+37\left(-\frac{1}{48}\right)=27$$

Multiply 37 times -\frac{1}{48}.

$$43y-\frac{37}{48}=27$$

Add \frac{37}{48} to both sides of the equation.

$$43y=\frac{1333}{48}$$

Divide both sides by 43.

$$y=\frac{31}{48}$$

The system is now solved.

$$y=\frac{31}{48},x=-\frac{1}{48}$$