Solve {l}{3x-3y=100}{5x+6y=460}

$$\left\{ \begin{array} { l } { 3 x - 3 y = 100 } \\ { 5 x + 6 y = 460 } \end{array} \right.$$

Solve for x, y

$$x=60$$

$$y = \frac{80}{3} = 26\frac{2}{3} \approx 26.666666667$$

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.

$$3x-3y=100,5x+6y=460$$

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

$$3x-3y=100$$

Add 3y to both sides of the equation.

$$3x=3y+100$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(3y+100\right)$$

Multiply \frac{1}{3} times 3y+100.

$$x=y+\frac{100}{3}$$

Substitute y+\frac{100}{3} for x in the other equation, 5x+6y=460.

$$5\left(y+\frac{100}{3}\right)+6y=460$$

Multiply 5 times y+\frac{100}{3}.

$$5y+\frac{500}{3}+6y=460$$

Add 5y to 6y.

$$11y+\frac{500}{3}=460$$

Subtract \frac{500}{3} from both sides of the equation.

$$11y=\frac{880}{3}$$

Divide both sides by 11.

$$y=\frac{80}{3}$$

Substitute \frac{80}{3} for y in x=y+\frac{100}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{80+100}{3}$$

Add \frac{100}{3} to \frac{80}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=60$$

The system is now solved.

$$x=60,y=\frac{80}{3}$$

Steps Using Matrices

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

$$3x-3y=100,5x+6y=460$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-3\\5&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\460\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-3\\5&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-3\\5&6\end{matrix}\right))\left(\begin{matrix}3&-3\\5&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-3\\5&6\end{matrix}\right))\left(\begin{matrix}100\\460\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-3\\5&6\end{matrix}\right))\left(\begin{matrix}100\\460\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-3\\5&6\end{matrix}\right))\left(\begin{matrix}100\\460\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{3\times 6-\left(-3\times 5\right)}&-\frac{-3}{3\times 6-\left(-3\times 5\right)}\\-\frac{5}{3\times 6-\left(-3\times 5\right)}&\frac{3}{3\times 6-\left(-3\times 5\right)}\end{matrix}\right)\left(\begin{matrix}100\\460\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&\frac{1}{11}\\-\frac{5}{33}&\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}100\\460\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 100+\frac{1}{11}\times 460\\-\frac{5}{33}\times 100+\frac{1}{11}\times 460\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\\frac{80}{3}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=60,y=\frac{80}{3}$$

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.

$$3x-3y=100,5x+6y=460$$

To make 3x 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 3.

$$5\times 3x+5\left(-3\right)y=5\times 100,3\times 5x+3\times 6y=3\times 460$$

Simplify.

$$15x-15y=500,15x+18y=1380$$

Subtract 15x+18y=1380 from 15x-15y=500 by subtracting like terms on each side of the equal sign.

$$15x-15x-15y-18y=500-1380$$

Add 15x to -15x. Terms 15x and -15x cancel out, leaving an equation with only one variable that can be solved.

$$-15y-18y=500-1380$$

Add -15y to -18y.

$$-33y=500-1380$$

Add 500 to -1380.

$$-33y=-880$$

Divide both sides by -33.

$$y=\frac{80}{3}$$

Substitute \frac{80}{3} for y in 5x+6y=460. Because the resulting equation contains only one variable, you can solve for x directly.

$$5x+6\times \frac{80}{3}=460$$

Multiply 6 times \frac{80}{3}.

$$5x+160=460$$

Subtract 160 from both sides of the equation.

$$5x=300$$

Divide both sides by 5.

$$x=60$$

The system is now solved.

$$x=60,y=\frac{80}{3}$$