Solve {l}{56x+21y=37}{34x-15y=18}

$$\left. \begin{array} { l } { 56 x + 21 y = 37 } \\ { 34 x - 15 y = 18 } \end{array} \right.$$

Solve for x, y

$$x=\frac{311}{518}\approx 0.6003861$$

$$y=\frac{125}{777}\approx 0.160875161$$

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.

$$56x+21y=37,34x-15y=18$$

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

$$56x+21y=37$$

Subtract 21y from both sides of the equation.

$$56x=-21y+37$$

Divide both sides by 56.

$$x=\frac{1}{56}\left(-21y+37\right)$$

Multiply \frac{1}{56} times -21y+37.

$$x=-\frac{3}{8}y+\frac{37}{56}$$

Substitute -\frac{3y}{8}+\frac{37}{56} for x in the other equation, 34x-15y=18.

$$34\left(-\frac{3}{8}y+\frac{37}{56}\right)-15y=18$$

Multiply 34 times -\frac{3y}{8}+\frac{37}{56}.

$$-\frac{51}{4}y+\frac{629}{28}-15y=18$$

Add -\frac{51y}{4} to -15y.

$$-\frac{111}{4}y+\frac{629}{28}=18$$

Subtract \frac{629}{28} from both sides of the equation.

$$-\frac{111}{4}y=-\frac{125}{28}$$

Divide both sides of the equation by -\frac{111}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{125}{777}$$

Substitute \frac{125}{777} for y in x=-\frac{3}{8}y+\frac{37}{56}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{8}\times \frac{125}{777}+\frac{37}{56}$$

Multiply -\frac{3}{8} times \frac{125}{777} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{125}{2072}+\frac{37}{56}$$

Add \frac{37}{56} to -\frac{125}{2072} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{311}{518}$$

The system is now solved.

$$x=\frac{311}{518},y=\frac{125}{777}$$

Steps Using Matrices

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

$$56x+21y=37,34x-15y=18$$

Write the equations in matrix form.

$$\left(\begin{matrix}56&21\\34&-15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}37\\18\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}56&21\\34&-15\end{matrix}\right).

$$inverse(\left(\begin{matrix}56&21\\34&-15\end{matrix}\right))\left(\begin{matrix}56&21\\34&-15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}56&21\\34&-15\end{matrix}\right))\left(\begin{matrix}37\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}56&21\\34&-15\end{matrix}\right))\left(\begin{matrix}37\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}56&21\\34&-15\end{matrix}\right))\left(\begin{matrix}37\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{56\left(-15\right)-21\times 34}&-\frac{21}{56\left(-15\right)-21\times 34}\\-\frac{34}{56\left(-15\right)-21\times 34}&\frac{56}{56\left(-15\right)-21\times 34}\end{matrix}\right)\left(\begin{matrix}37\\18\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{518}&\frac{1}{74}\\\frac{17}{777}&-\frac{4}{111}\end{matrix}\right)\left(\begin{matrix}37\\18\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{518}\times 37+\frac{1}{74}\times 18\\\frac{17}{777}\times 37-\frac{4}{111}\times 18\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{311}{518}\\\frac{125}{777}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{311}{518},y=\frac{125}{777}$$

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.

$$56x+21y=37,34x-15y=18$$

To make 56x and 34x equal, multiply all terms on each side of the first equation by 34 and all terms on each side of the second by 56.

$$34\times 56x+34\times 21y=34\times 37,56\times 34x+56\left(-15\right)y=56\times 18$$

Simplify.

$$1904x+714y=1258,1904x-840y=1008$$

Subtract 1904x-840y=1008 from 1904x+714y=1258 by subtracting like terms on each side of the equal sign.

$$1904x-1904x+714y+840y=1258-1008$$

Add 1904x to -1904x. Terms 1904x and -1904x cancel out, leaving an equation with only one variable that can be solved.

$$714y+840y=1258-1008$$

Add 714y to 840y.

$$1554y=1258-1008$$

Add 1258 to -1008.

$$1554y=250$$

Divide both sides by 1554.

$$y=\frac{125}{777}$$

Substitute \frac{125}{777} for y in 34x-15y=18. Because the resulting equation contains only one variable, you can solve for x directly.

$$34x-15\times \frac{125}{777}=18$$

Multiply -15 times \frac{125}{777}.

$$34x-\frac{625}{259}=18$$

Add \frac{625}{259} to both sides of the equation.

$$34x=\frac{5287}{259}$$

Divide both sides by 34.

$$x=\frac{311}{518}$$

The system is now solved.

$$x=\frac{311}{518},y=\frac{125}{777}$$