Solve {l}{6x+5y=1380}{10x+7y=20600}

$$\left\{ \begin{array} { l } { 6 x + 5 y = 1380 } \\ { 10 x + 7 y = 20600 } \end{array} \right.$$

Solve for x, y

$$x = \frac{23335}{2} = 11667\frac{1}{2} = 11667.5$$

$$y=-13725$$

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.

$$6x+5y=1380,10x+7y=20600$$

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

$$6x+5y=1380$$

Subtract 5y from both sides of the equation.

$$6x=-5y+1380$$

Divide both sides by 6.

$$x=\frac{1}{6}\left(-5y+1380\right)$$

Multiply \frac{1}{6} times -5y+1380.

$$x=-\frac{5}{6}y+230$$

Substitute -\frac{5y}{6}+230 for x in the other equation, 10x+7y=20600.

$$10\left(-\frac{5}{6}y+230\right)+7y=20600$$

Multiply 10 times -\frac{5y}{6}+230.

$$-\frac{25}{3}y+2300+7y=20600$$

Add -\frac{25y}{3} to 7y.

$$-\frac{4}{3}y+2300=20600$$

Subtract 2300 from both sides of the equation.

$$-\frac{4}{3}y=18300$$

Divide both sides of the equation by -\frac{4}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-13725$$

Substitute -13725 for y in x=-\frac{5}{6}y+230. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{5}{6}\left(-13725\right)+230$$

Multiply -\frac{5}{6} times -13725.

$$x=\frac{22875}{2}+230$$

Add 230 to \frac{22875}{2}.

$$x=\frac{23335}{2}$$

The system is now solved.

$$x=\frac{23335}{2},y=-13725$$

Steps Using Matrices

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

$$6x+5y=1380,10x+7y=20600$$

Write the equations in matrix form.

$$\left(\begin{matrix}6&5\\10&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1380\\20600\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}6&5\\10&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&5\\10&7\end{matrix}\right))\left(\begin{matrix}6&5\\10&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\10&7\end{matrix}\right))\left(\begin{matrix}1380\\20600\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\10&7\end{matrix}\right))\left(\begin{matrix}1380\\20600\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\10&7\end{matrix}\right))\left(\begin{matrix}1380\\20600\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{6\times 7-5\times 10}&-\frac{5}{6\times 7-5\times 10}\\-\frac{10}{6\times 7-5\times 10}&\frac{6}{6\times 7-5\times 10}\end{matrix}\right)\left(\begin{matrix}1380\\20600\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{8}&\frac{5}{8}\\\frac{5}{4}&-\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}1380\\20600\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{8}\times 1380+\frac{5}{8}\times 20600\\\frac{5}{4}\times 1380-\frac{3}{4}\times 20600\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23335}{2}\\-13725\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{23335}{2},y=-13725$$

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.

$$6x+5y=1380,10x+7y=20600$$

To make 6x and 10x equal, multiply all terms on each side of the first equation by 10 and all terms on each side of the second by 6.

$$10\times 6x+10\times 5y=10\times 1380,6\times 10x+6\times 7y=6\times 20600$$

Simplify.

$$60x+50y=13800,60x+42y=123600$$

Subtract 60x+42y=123600 from 60x+50y=13800 by subtracting like terms on each side of the equal sign.

$$60x-60x+50y-42y=13800-123600$$

Add 60x to -60x. Terms 60x and -60x cancel out, leaving an equation with only one variable that can be solved.

$$50y-42y=13800-123600$$

Add 50y to -42y.

$$8y=13800-123600$$

Add 13800 to -123600.

$$8y=-109800$$

Divide both sides by 8.

$$y=-13725$$

Substitute -13725 for y in 10x+7y=20600. Because the resulting equation contains only one variable, you can solve for x directly.

$$10x+7\left(-13725\right)=20600$$

Multiply 7 times -13725.

$$10x-96075=20600$$

Add 96075 to both sides of the equation.

$$10x=116675$$

Divide both sides by 10.

$$x=\frac{23335}{2}$$

The system is now solved.

$$x=\frac{23335}{2},y=-13725$$