Solve {l}{x+y=200}{87x+43y=15200}

$$\left\{ \begin{array} { l } { x + y = 200 } \\ { 87 x + 43 y = 15200 } \end{array} \right.$$

Solve for x, y

$$x=150$$

$$y=50$$

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.

$$x+y=200,87x+43y=15200$$

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

$$x+y=200$$

Subtract y from both sides of the equation.

$$x=-y+200$$

Substitute -y+200 for x in the other equation, 87x+43y=15200.

$$87\left(-y+200\right)+43y=15200$$

Multiply 87 times -y+200.

$$-87y+17400+43y=15200$$

Add -87y to 43y.

$$-44y+17400=15200$$

Subtract 17400 from both sides of the equation.

$$-44y=-2200$$

Divide both sides by -44.

$$y=50$$

Substitute 50 for y in x=-y+200. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-50+200$$

Add 200 to -50.

$$x=150$$

The system is now solved.

$$x=150,y=50$$

Steps Using Matrices

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

$$x+y=200,87x+43y=15200$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\87&43\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}200\\15200\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\87&43\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\87&43\end{matrix}\right))\left(\begin{matrix}1&1\\87&43\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\87&43\end{matrix}\right))\left(\begin{matrix}200\\15200\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\87&43\end{matrix}\right))\left(\begin{matrix}200\\15200\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\87&43\end{matrix}\right))\left(\begin{matrix}200\\15200\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{43}{43-87}&-\frac{1}{43-87}\\-\frac{87}{43-87}&\frac{1}{43-87}\end{matrix}\right)\left(\begin{matrix}200\\15200\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{43}{44}&\frac{1}{44}\\\frac{87}{44}&-\frac{1}{44}\end{matrix}\right)\left(\begin{matrix}200\\15200\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{43}{44}\times 200+\frac{1}{44}\times 15200\\\frac{87}{44}\times 200-\frac{1}{44}\times 15200\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}150\\50\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=150,y=50$$

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.

$$x+y=200,87x+43y=15200$$

To make x and 87x equal, multiply all terms on each side of the first equation by 87 and all terms on each side of the second by 1.

$$87x+87y=87\times 200,87x+43y=15200$$

Simplify.

$$87x+87y=17400,87x+43y=15200$$

Subtract 87x+43y=15200 from 87x+87y=17400 by subtracting like terms on each side of the equal sign.

$$87x-87x+87y-43y=17400-15200$$

Add 87x to -87x. Terms 87x and -87x cancel out, leaving an equation with only one variable that can be solved.

$$87y-43y=17400-15200$$

Add 87y to -43y.

$$44y=17400-15200$$

Add 17400 to -15200.

$$44y=2200$$

Divide both sides by 44.

$$y=50$$

Substitute 50 for y in 87x+43y=15200. Because the resulting equation contains only one variable, you can solve for x directly.

$$87x+43\times 50=15200$$

Multiply 43 times 50.

$$87x+2150=15200$$

Subtract 2150 from both sides of the equation.

$$87x=13050$$

Divide both sides by 87.

$$x=150$$

The system is now solved.

$$x=150,y=50$$