Solve {l}{11x+2y=27}{21x+3y=-9}

$$\left\{ \begin{array} { l } { 11 x + 2 y = 27 } \\ { 21 x + 3 y = - 9 } \end{array} \right.$$

Solve for x, y

$$x=-11$$

$$y=74$$

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.

$$11x+2y=27,21x+3y=-9$$

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

$$11x+2y=27$$

Subtract 2y from both sides of the equation.

$$11x=-2y+27$$

Divide both sides by 11.

$$x=\frac{1}{11}\left(-2y+27\right)$$

Multiply \frac{1}{11} times -2y+27.

$$x=-\frac{2}{11}y+\frac{27}{11}$$

Substitute \frac{-2y+27}{11} for x in the other equation, 21x+3y=-9.

$$21\left(-\frac{2}{11}y+\frac{27}{11}\right)+3y=-9$$

Multiply 21 times \frac{-2y+27}{11}.

$$-\frac{42}{11}y+\frac{567}{11}+3y=-9$$

Add -\frac{42y}{11} to 3y.

$$-\frac{9}{11}y+\frac{567}{11}=-9$$

Subtract \frac{567}{11} from both sides of the equation.

$$-\frac{9}{11}y=-\frac{666}{11}$$

Divide both sides of the equation by -\frac{9}{11}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=74$$

Substitute 74 for y in x=-\frac{2}{11}y+\frac{27}{11}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{2}{11}\times 74+\frac{27}{11}$$

Multiply -\frac{2}{11} times 74.

$$x=\frac{-148+27}{11}$$

Add \frac{27}{11} to -\frac{148}{11} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-11$$

The system is now solved.

$$x=-11,y=74$$

Steps Using Matrices

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

$$11x+2y=27,21x+3y=-9$$

Write the equations in matrix form.

$$\left(\begin{matrix}11&2\\21&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\-9\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}11&2\\21&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}11&2\\21&3\end{matrix}\right))\left(\begin{matrix}11&2\\21&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&2\\21&3\end{matrix}\right))\left(\begin{matrix}27\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&2\\21&3\end{matrix}\right))\left(\begin{matrix}27\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&2\\21&3\end{matrix}\right))\left(\begin{matrix}27\\-9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11\times 3-2\times 21}&-\frac{2}{11\times 3-2\times 21}\\-\frac{21}{11\times 3-2\times 21}&\frac{11}{11\times 3-2\times 21}\end{matrix}\right)\left(\begin{matrix}27\\-9\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{2}{9}\\\frac{7}{3}&-\frac{11}{9}\end{matrix}\right)\left(\begin{matrix}27\\-9\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 27+\frac{2}{9}\left(-9\right)\\\frac{7}{3}\times 27-\frac{11}{9}\left(-9\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11\\74\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-11,y=74$$

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.

$$11x+2y=27,21x+3y=-9$$

To make 11x and 21x equal, multiply all terms on each side of the first equation by 21 and all terms on each side of the second by 11.

$$21\times 11x+21\times 2y=21\times 27,11\times 21x+11\times 3y=11\left(-9\right)$$

Simplify.

$$231x+42y=567,231x+33y=-99$$

Subtract 231x+33y=-99 from 231x+42y=567 by subtracting like terms on each side of the equal sign.

$$231x-231x+42y-33y=567+99$$

Add 231x to -231x. Terms 231x and -231x cancel out, leaving an equation with only one variable that can be solved.

$$42y-33y=567+99$$

Add 42y to -33y.

$$9y=567+99$$

Add 567 to 99.

$$9y=666$$

Divide both sides by 9.

$$y=74$$

Substitute 74 for y in 21x+3y=-9. Because the resulting equation contains only one variable, you can solve for x directly.

$$21x+3\times 74=-9$$

Multiply 3 times 74.

$$21x+222=-9$$

Subtract 222 from both sides of the equation.

$$21x=-231$$

Divide both sides by 21.

$$x=-11$$

The system is now solved.

$$x=-11,y=74$$