Solve {l}{9x+18y=900}{15x+6y=780}

$$\left. \begin{array} { l } { 9 x + 18 y = 900 } \\ { 15 x + 6 y = 780 } \end{array} \right.$$

Solve for x, y

$$x=40$$

$$y=30$$

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.

$$9x+18y=900,15x+6y=780$$

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

$$9x+18y=900$$

Subtract 18y from both sides of the equation.

$$9x=-18y+900$$

Divide both sides by 9.

$$x=\frac{1}{9}\left(-18y+900\right)$$

Multiply \frac{1}{9} times -18y+900.

$$x=-2y+100$$

Substitute -2y+100 for x in the other equation, 15x+6y=780.

$$15\left(-2y+100\right)+6y=780$$

Multiply 15 times -2y+100.

$$-30y+1500+6y=780$$

Add -30y to 6y.

$$-24y+1500=780$$

Subtract 1500 from both sides of the equation.

$$-24y=-720$$

Divide both sides by -24.

$$y=30$$

Substitute 30 for y in x=-2y+100. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-2\times 30+100$$

Multiply -2 times 30.

$$x=-60+100$$

Add 100 to -60.

$$x=40$$

The system is now solved.

$$x=40,y=30$$

Steps Using Matrices

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

$$9x+18y=900,15x+6y=780$$

Write the equations in matrix form.

$$\left(\begin{matrix}9&18\\15&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}900\\780\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}9&18\\15&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&18\\15&6\end{matrix}\right))\left(\begin{matrix}9&18\\15&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&18\\15&6\end{matrix}\right))\left(\begin{matrix}900\\780\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&18\\15&6\end{matrix}\right))\left(\begin{matrix}900\\780\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&18\\15&6\end{matrix}\right))\left(\begin{matrix}900\\780\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{9\times 6-18\times 15}&-\frac{18}{9\times 6-18\times 15}\\-\frac{15}{9\times 6-18\times 15}&\frac{9}{9\times 6-18\times 15}\end{matrix}\right)\left(\begin{matrix}900\\780\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{36}&\frac{1}{12}\\\frac{5}{72}&-\frac{1}{24}\end{matrix}\right)\left(\begin{matrix}900\\780\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{36}\times 900+\frac{1}{12}\times 780\\\frac{5}{72}\times 900-\frac{1}{24}\times 780\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\30\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=40,y=30$$

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.

$$9x+18y=900,15x+6y=780$$

To make 9x and 15x equal, multiply all terms on each side of the first equation by 15 and all terms on each side of the second by 9.

$$15\times 9x+15\times 18y=15\times 900,9\times 15x+9\times 6y=9\times 780$$

Simplify.

$$135x+270y=13500,135x+54y=7020$$

Subtract 135x+54y=7020 from 135x+270y=13500 by subtracting like terms on each side of the equal sign.

$$135x-135x+270y-54y=13500-7020$$

Add 135x to -135x. Terms 135x and -135x cancel out, leaving an equation with only one variable that can be solved.

$$270y-54y=13500-7020$$

Add 270y to -54y.

$$216y=13500-7020$$

Add 13500 to -7020.

$$216y=6480$$

Divide both sides by 216.

$$y=30$$

Substitute 30 for y in 15x+6y=780. Because the resulting equation contains only one variable, you can solve for x directly.

$$15x+6\times 30=780$$

Multiply 6 times 30.

$$15x+180=780$$

Subtract 180 from both sides of the equation.

$$15x=600$$

Divide both sides by 15.

$$x=40$$

The system is now solved.

$$x=40,y=30$$