Solve {l}{4x+3y=140000}{5x+4y=180000}

$$\left. \begin{array} { l } { 4 x + 3 y = 140000 } \\ { 5 x + 4 y = 180000 } \end{array} \right.$$

Solve for x, y

$$x=20000$$

$$y=20000$$

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.

$$4x+3y=140000,5x+4y=180000$$

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

$$4x+3y=140000$$

Subtract 3y from both sides of the equation.

$$4x=-3y+140000$$

Divide both sides by 4.

$$x=\frac{1}{4}\left(-3y+140000\right)$$

Multiply \frac{1}{4} times -3y+140000.

$$x=-\frac{3}{4}y+35000$$

Substitute -\frac{3y}{4}+35000 for x in the other equation, 5x+4y=180000.

$$5\left(-\frac{3}{4}y+35000\right)+4y=180000$$

Multiply 5 times -\frac{3y}{4}+35000.

$$-\frac{15}{4}y+175000+4y=180000$$

Add -\frac{15y}{4} to 4y.

$$\frac{1}{4}y+175000=180000$$

Subtract 175000 from both sides of the equation.

$$\frac{1}{4}y=5000$$

Multiply both sides by 4.

$$y=20000$$

Substitute 20000 for y in x=-\frac{3}{4}y+35000. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{4}\times 20000+35000$$

Multiply -\frac{3}{4} times 20000.

$$x=-15000+35000$$

Add 35000 to -15000.

$$x=20000$$

The system is now solved.

$$x=20000,y=20000$$

Steps Using Matrices

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

$$4x+3y=140000,5x+4y=180000$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&3\\5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}140000\\180000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&3\\5&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&3\\5&4\end{matrix}\right))\left(\begin{matrix}4&3\\5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\5&4\end{matrix}\right))\left(\begin{matrix}140000\\180000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\5&4\end{matrix}\right))\left(\begin{matrix}140000\\180000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\5&4\end{matrix}\right))\left(\begin{matrix}140000\\180000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4\times 4-3\times 5}&-\frac{3}{4\times 4-3\times 5}\\-\frac{5}{4\times 4-3\times 5}&\frac{4}{4\times 4-3\times 5}\end{matrix}\right)\left(\begin{matrix}140000\\180000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4&-3\\-5&4\end{matrix}\right)\left(\begin{matrix}140000\\180000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\times 140000-3\times 180000\\-5\times 140000+4\times 180000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20000\\20000\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=20000,y=20000$$

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.

$$4x+3y=140000,5x+4y=180000$$

To make 4x and 5x equal, multiply all terms on each side of the first equation by 5 and all terms on each side of the second by 4.

$$5\times 4x+5\times 3y=5\times 140000,4\times 5x+4\times 4y=4\times 180000$$

Simplify.

$$20x+15y=700000,20x+16y=720000$$

Subtract 20x+16y=720000 from 20x+15y=700000 by subtracting like terms on each side of the equal sign.

$$20x-20x+15y-16y=700000-720000$$

Add 20x to -20x. Terms 20x and -20x cancel out, leaving an equation with only one variable that can be solved.

$$15y-16y=700000-720000$$

Add 15y to -16y.

$$-y=700000-720000$$

Add 700000 to -720000.

$$-y=-20000$$

Divide both sides by -1.

$$y=20000$$

Substitute 20000 for y in 5x+4y=180000. Because the resulting equation contains only one variable, you can solve for x directly.

$$5x+4\times 20000=180000$$

Multiply 4 times 20000.

$$5x+80000=180000$$

Subtract 80000 from both sides of the equation.

$$5x=100000$$

Divide both sides by 5.

$$x=20000$$

The system is now solved.

$$x=20000,y=20000$$