Solve {l}{20x+30y=12000}{25x+36y=14700}

$$\left. \begin{array} { l } { 20 x + 30 y = 12000 } \\ { 25 x + 36 y = 14700 } \end{array} \right.$$

Solve for x, y

$$x=300$$

$$y=200$$

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.

$$20x+30y=12000,25x+36y=14700$$

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

$$20x+30y=12000$$

Subtract 30y from both sides of the equation.

$$20x=-30y+12000$$

Divide both sides by 20.

$$x=\frac{1}{20}\left(-30y+12000\right)$$

Multiply \frac{1}{20} times -30y+12000.

$$x=-\frac{3}{2}y+600$$

Substitute -\frac{3y}{2}+600 for x in the other equation, 25x+36y=14700.

$$25\left(-\frac{3}{2}y+600\right)+36y=14700$$

Multiply 25 times -\frac{3y}{2}+600.

$$-\frac{75}{2}y+15000+36y=14700$$

Add -\frac{75y}{2} to 36y.

$$-\frac{3}{2}y+15000=14700$$

Subtract 15000 from both sides of the equation.

$$-\frac{3}{2}y=-300$$

Divide both sides of the equation by -\frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=200$$

Substitute 200 for y in x=-\frac{3}{2}y+600. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{2}\times 200+600$$

Multiply -\frac{3}{2} times 200.

$$x=-300+600$$

Add 600 to -300.

$$x=300$$

The system is now solved.

$$x=300,y=200$$

Steps Using Matrices

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

$$20x+30y=12000,25x+36y=14700$$

Write the equations in matrix form.

$$\left(\begin{matrix}20&30\\25&36\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12000\\14700\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}20&30\\25&36\end{matrix}\right).

$$inverse(\left(\begin{matrix}20&30\\25&36\end{matrix}\right))\left(\begin{matrix}20&30\\25&36\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&30\\25&36\end{matrix}\right))\left(\begin{matrix}12000\\14700\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&30\\25&36\end{matrix}\right))\left(\begin{matrix}12000\\14700\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&30\\25&36\end{matrix}\right))\left(\begin{matrix}12000\\14700\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{36}{20\times 36-30\times 25}&-\frac{30}{20\times 36-30\times 25}\\-\frac{25}{20\times 36-30\times 25}&\frac{20}{20\times 36-30\times 25}\end{matrix}\right)\left(\begin{matrix}12000\\14700\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{5}&1\\\frac{5}{6}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}12000\\14700\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{5}\times 12000+14700\\\frac{5}{6}\times 12000-\frac{2}{3}\times 14700\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}300\\200\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=300,y=200$$

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.

$$20x+30y=12000,25x+36y=14700$$

To make 20x and 25x equal, multiply all terms on each side of the first equation by 25 and all terms on each side of the second by 20.

$$25\times 20x+25\times 30y=25\times 12000,20\times 25x+20\times 36y=20\times 14700$$

Simplify.

$$500x+750y=300000,500x+720y=294000$$

Subtract 500x+720y=294000 from 500x+750y=300000 by subtracting like terms on each side of the equal sign.

$$500x-500x+750y-720y=300000-294000$$

Add 500x to -500x. Terms 500x and -500x cancel out, leaving an equation with only one variable that can be solved.

$$750y-720y=300000-294000$$

Add 750y to -720y.

$$30y=300000-294000$$

Add 300000 to -294000.

$$30y=6000$$

Divide both sides by 30.

$$y=200$$

Substitute 200 for y in 25x+36y=14700. Because the resulting equation contains only one variable, you can solve for x directly.

$$25x+36\times 200=14700$$

Multiply 36 times 200.

$$25x+7200=14700$$

Subtract 7200 from both sides of the equation.

$$25x=7500$$

Divide both sides by 25.

$$x=300$$

The system is now solved.

$$x=300,y=200$$