Solve {l}{-6x-27=-9y}{7x-7y=7}

$$\left. \begin{array} { l } { - 6 x - 27 = - 9 y } \\ { 7 x - 7 y = 7 } \end{array} \right.$$

Solve for x, y

$$x=12$$

$$y=11$$

Steps Using Substitution

Consider the first equation. Add 9y to both sides.

$$-6x-27+9y=0$$

Add 27 to both sides. Anything plus zero gives itself.

$$-6x+9y=27$$

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.

$$-6x+9y=27,7x-7y=7$$

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

$$-6x+9y=27$$

Subtract 9y from both sides of the equation.

$$-6x=-9y+27$$

Divide both sides by -6.

$$x=-\frac{1}{6}\left(-9y+27\right)$$

Multiply -\frac{1}{6} times -9y+27.

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

Substitute \frac{-9+3y}{2} for x in the other equation, 7x-7y=7.

$$7\left(\frac{3}{2}y-\frac{9}{2}\right)-7y=7$$

Multiply 7 times \frac{-9+3y}{2}.

$$\frac{21}{2}y-\frac{63}{2}-7y=7$$

Add \frac{21y}{2} to -7y.

$$\frac{7}{2}y-\frac{63}{2}=7$$

Add \frac{63}{2} to both sides of the equation.

$$\frac{7}{2}y=\frac{77}{2}$$

Divide both sides of the equation by \frac{7}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=11$$

Substitute 11 for y in x=\frac{3}{2}y-\frac{9}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{3}{2}\times 11-\frac{9}{2}$$

Multiply \frac{3}{2} times 11.

$$x=\frac{33-9}{2}$$

Add -\frac{9}{2} to \frac{33}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=12$$

The system is now solved.

$$x=12,y=11$$

Steps Using Matrices

Consider the first equation. Add 9y to both sides.

$$-6x-27+9y=0$$

Add 27 to both sides. Anything plus zero gives itself.

$$-6x+9y=27$$

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

$$-6x+9y=27,7x-7y=7$$

Write the equations in matrix form.

$$\left(\begin{matrix}-6&9\\7&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\7\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-6&9\\7&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&9\\7&-7\end{matrix}\right))\left(\begin{matrix}-6&9\\7&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&9\\7&-7\end{matrix}\right))\left(\begin{matrix}27\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&9\\7&-7\end{matrix}\right))\left(\begin{matrix}27\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&9\\7&-7\end{matrix}\right))\left(\begin{matrix}27\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{-6\left(-7\right)-9\times 7}&-\frac{9}{-6\left(-7\right)-9\times 7}\\-\frac{7}{-6\left(-7\right)-9\times 7}&-\frac{6}{-6\left(-7\right)-9\times 7}\end{matrix}\right)\left(\begin{matrix}27\\7\end{matrix}\right)$$

Do the arithmetic.

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

Multiply the matrices.

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

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\11\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=12,y=11$$

Steps Using Elimination

Consider the first equation. Add 9y to both sides.

$$-6x-27+9y=0$$

Add 27 to both sides. Anything plus zero gives itself.

$$-6x+9y=27$$

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.

$$-6x+9y=27,7x-7y=7$$

To make -6x and 7x equal, multiply all terms on each side of the first equation by 7 and all terms on each side of the second by -6.

$$7\left(-6\right)x+7\times 9y=7\times 27,-6\times 7x-6\left(-7\right)y=-6\times 7$$

Simplify.

$$-42x+63y=189,-42x+42y=-42$$

Subtract -42x+42y=-42 from -42x+63y=189 by subtracting like terms on each side of the equal sign.

$$-42x+42x+63y-42y=189+42$$

Add -42x to 42x. Terms -42x and 42x cancel out, leaving an equation with only one variable that can be solved.

$$63y-42y=189+42$$

Add 63y to -42y.

$$21y=189+42$$

Add 189 to 42.

$$21y=231$$

Divide both sides by 21.

$$y=11$$

Substitute 11 for y in 7x-7y=7. Because the resulting equation contains only one variable, you can solve for x directly.

$$7x-7\times 11=7$$

Multiply -7 times 11.

$$7x-77=7$$

Add 77 to both sides of the equation.

$$7x=84$$

Divide both sides by 7.

$$x=12$$

The system is now solved.

$$x=12,y=11$$