Solve {l}{10x+20y=650}{7x+9y=355}

$$\left\{ \begin{array} { l } { 10 x + 20 y = 650 } \\ { 7 x + 9 y = 355 } \end{array} \right.$$

Solve for x, y

$$x=25$$

$$y=20$$

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.

$$10x+20y=650,7x+9y=355$$

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

$$10x+20y=650$$

Subtract 20y from both sides of the equation.

$$10x=-20y+650$$

Divide both sides by 10.

$$x=\frac{1}{10}\left(-20y+650\right)$$

Multiply \frac{1}{10} times -20y+650.

$$x=-2y+65$$

Substitute -2y+65 for x in the other equation, 7x+9y=355.

$$7\left(-2y+65\right)+9y=355$$

Multiply 7 times -2y+65.

$$-14y+455+9y=355$$

Add -14y to 9y.

$$-5y+455=355$$

Subtract 455 from both sides of the equation.

$$-5y=-100$$

Divide both sides by -5.

$$y=20$$

Substitute 20 for y in x=-2y+65. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-2\times 20+65$$

Multiply -2 times 20.

$$x=-40+65$$

Add 65 to -40.

$$x=25$$

The system is now solved.

$$x=25,y=20$$

Steps Using Matrices

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

$$10x+20y=650,7x+9y=355$$

Write the equations in matrix form.

$$\left(\begin{matrix}10&20\\7&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}650\\355\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}10&20\\7&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&20\\7&9\end{matrix}\right))\left(\begin{matrix}10&20\\7&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&20\\7&9\end{matrix}\right))\left(\begin{matrix}650\\355\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&20\\7&9\end{matrix}\right))\left(\begin{matrix}650\\355\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&20\\7&9\end{matrix}\right))\left(\begin{matrix}650\\355\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{10\times 9-20\times 7}&-\frac{20}{10\times 9-20\times 7}\\-\frac{7}{10\times 9-20\times 7}&\frac{10}{10\times 9-20\times 7}\end{matrix}\right)\left(\begin{matrix}650\\355\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{50}&\frac{2}{5}\\\frac{7}{50}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}650\\355\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{50}\times 650+\frac{2}{5}\times 355\\\frac{7}{50}\times 650-\frac{1}{5}\times 355\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\20\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=25,y=20$$

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.

$$10x+20y=650,7x+9y=355$$

To make 10x and 7x equal, multiply all terms on each side of the first equation by 7 and all terms on each side of the second by 10.

$$7\times 10x+7\times 20y=7\times 650,10\times 7x+10\times 9y=10\times 355$$

Simplify.

$$70x+140y=4550,70x+90y=3550$$

Subtract 70x+90y=3550 from 70x+140y=4550 by subtracting like terms on each side of the equal sign.

$$70x-70x+140y-90y=4550-3550$$

Add 70x to -70x. Terms 70x and -70x cancel out, leaving an equation with only one variable that can be solved.

$$140y-90y=4550-3550$$

Add 140y to -90y.

$$50y=4550-3550$$

Add 4550 to -3550.

$$50y=1000$$

Divide both sides by 50.

$$y=20$$

Substitute 20 for y in 7x+9y=355. Because the resulting equation contains only one variable, you can solve for x directly.

$$7x+9\times 20=355$$

Multiply 9 times 20.

$$7x+180=355$$

Subtract 180 from both sides of the equation.

$$7x=175$$

Divide both sides by 7.

$$x=25$$

The system is now solved.

$$x=25,y=20$$