Solve {l}{200x+300y=360}{300x+200y=340}

$$\left\{ \begin{array} { l } { 200 x + 300 y = 360 } \\ { 300 x + 200 y = 340 } \end{array} \right.$$

Solve for x, y

$$x=\frac{3}{5}=0.6$$

$$y=\frac{4}{5}=0.8$$

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.

$$200x+300y=360,300x+200y=340$$

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

$$200x+300y=360$$

Subtract 300y from both sides of the equation.

$$200x=-300y+360$$

Divide both sides by 200.

$$x=\frac{1}{200}\left(-300y+360\right)$$

Multiply \frac{1}{200} times -300y+360.

$$x=-\frac{3}{2}y+\frac{9}{5}$$

Substitute -\frac{3y}{2}+\frac{9}{5} for x in the other equation, 300x+200y=340.

$$300\left(-\frac{3}{2}y+\frac{9}{5}\right)+200y=340$$

Multiply 300 times -\frac{3y}{2}+\frac{9}{5}.

$$-450y+540+200y=340$$

Add -450y to 200y.

$$-250y+540=340$$

Subtract 540 from both sides of the equation.

$$-250y=-200$$

Divide both sides by -250.

$$y=\frac{4}{5}$$

Substitute \frac{4}{5} for y in x=-\frac{3}{2}y+\frac{9}{5}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{2}\times \frac{4}{5}+\frac{9}{5}$$

Multiply -\frac{3}{2} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-6+9}{5}$$

Add \frac{9}{5} to -\frac{6}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{3}{5}$$

The system is now solved.

$$x=\frac{3}{5},y=\frac{4}{5}$$

Steps Using Matrices

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

$$200x+300y=360,300x+200y=340$$

Write the equations in matrix form.

$$\left(\begin{matrix}200&300\\300&200\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}360\\340\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}200&300\\300&200\end{matrix}\right).

$$inverse(\left(\begin{matrix}200&300\\300&200\end{matrix}\right))\left(\begin{matrix}200&300\\300&200\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}200&300\\300&200\end{matrix}\right))\left(\begin{matrix}360\\340\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}200&300\\300&200\end{matrix}\right))\left(\begin{matrix}360\\340\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}200&300\\300&200\end{matrix}\right))\left(\begin{matrix}360\\340\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{200}{200\times 200-300\times 300}&-\frac{300}{200\times 200-300\times 300}\\-\frac{300}{200\times 200-300\times 300}&\frac{200}{200\times 200-300\times 300}\end{matrix}\right)\left(\begin{matrix}360\\340\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{250}&\frac{3}{500}\\\frac{3}{500}&-\frac{1}{250}\end{matrix}\right)\left(\begin{matrix}360\\340\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{250}\times 360+\frac{3}{500}\times 340\\\frac{3}{500}\times 360-\frac{1}{250}\times 340\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\\\frac{4}{5}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{3}{5},y=\frac{4}{5}$$

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.

$$200x+300y=360,300x+200y=340$$

To make 200x and 300x equal, multiply all terms on each side of the first equation by 300 and all terms on each side of the second by 200.

$$300\times 200x+300\times 300y=300\times 360,200\times 300x+200\times 200y=200\times 340$$

Simplify.

$$60000x+90000y=108000,60000x+40000y=68000$$

Subtract 60000x+40000y=68000 from 60000x+90000y=108000 by subtracting like terms on each side of the equal sign.

$$60000x-60000x+90000y-40000y=108000-68000$$

Add 60000x to -60000x. Terms 60000x and -60000x cancel out, leaving an equation with only one variable that can be solved.

$$90000y-40000y=108000-68000$$

Add 90000y to -40000y.

$$50000y=108000-68000$$

Add 108000 to -68000.

$$50000y=40000$$

Divide both sides by 50000.

$$y=\frac{4}{5}$$

Substitute \frac{4}{5} for y in 300x+200y=340. Because the resulting equation contains only one variable, you can solve for x directly.

$$300x+200\times \frac{4}{5}=340$$

Multiply 200 times \frac{4}{5}.

$$300x+160=340$$

Subtract 160 from both sides of the equation.

$$300x=180$$

Divide both sides by 300.

$$x=\frac{3}{5}$$

The system is now solved.

$$x=\frac{3}{5},y=\frac{4}{5}$$