Solve {l}{(3x-2)(2y+3)=(6x-1)*(y+3)-27}{(4x+1)*(3y+2)=(2x+5)(6y-4)}

$$\left\{ \begin{array} { l } { ( 3 x - 2 ) ( 2 y + 3 ) = ( 6 x - 1 ) \cdot ( y + 3 ) - 27 } \\ { ( 4 x + 1 ) \cdot ( 3 y + 2 ) = ( 2 x + 5 ) ( 6 y - 4 ) } \end{array} \right.$$

Solve for x, y

$$x=2$$

$$y=2$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 3x-2 by 2y+3.

$$6xy+9x-4y-6=\left(6x-1\right)\left(y+3\right)-27$$

Use the distributive property to multiply 6x-1 by y+3.

$$6xy+9x-4y-6=6xy+18x-y-3-27$$

Subtract 27 from -3 to get -30.

$$6xy+9x-4y-6=6xy+18x-y-30$$

Subtract 6xy from both sides.

$$6xy+9x-4y-6-6xy=18x-y-30$$

Combine 6xy and -6xy to get 0.

$$9x-4y-6=18x-y-30$$

Subtract 18x from both sides.

$$9x-4y-6-18x=-y-30$$

Combine 9x and -18x to get -9x.

$$-9x-4y-6=-y-30$$

Add y to both sides.

$$-9x-4y-6+y=-30$$

Combine -4y and y to get -3y.

$$-9x-3y-6=-30$$

Add 6 to both sides.

$$-9x-3y=-30+6$$

Add -30 and 6 to get -24.

$$-9x-3y=-24$$

Consider the second equation. Use the distributive property to multiply 4x+1 by 3y+2.

$$12xy+8x+3y+2=\left(2x+5\right)\left(6y-4\right)$$

Use the distributive property to multiply 2x+5 by 6y-4.

$$12xy+8x+3y+2=12xy-8x+30y-20$$

Subtract 12xy from both sides.

$$12xy+8x+3y+2-12xy=-8x+30y-20$$

Combine 12xy and -12xy to get 0.

$$8x+3y+2=-8x+30y-20$$

Add 8x to both sides.

$$8x+3y+2+8x=30y-20$$

Combine 8x and 8x to get 16x.

$$16x+3y+2=30y-20$$

Subtract 30y from both sides.

$$16x+3y+2-30y=-20$$

Combine 3y and -30y to get -27y.

$$16x-27y+2=-20$$

Subtract 2 from both sides.

$$16x-27y=-20-2$$

Subtract 2 from -20 to get -22.

$$16x-27y=-22$$

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.

$$-9x-3y=-24,16x-27y=-22$$

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

$$-9x-3y=-24$$

Add 3y to both sides of the equation.

$$-9x=3y-24$$

Divide both sides by -9.

$$x=-\frac{1}{9}\left(3y-24\right)$$

Multiply -\frac{1}{9} times -24+3y.

$$x=-\frac{1}{3}y+\frac{8}{3}$$

Substitute \frac{-y+8}{3} for x in the other equation, 16x-27y=-22.

$$16\left(-\frac{1}{3}y+\frac{8}{3}\right)-27y=-22$$

Multiply 16 times \frac{-y+8}{3}.

$$-\frac{16}{3}y+\frac{128}{3}-27y=-22$$

Add -\frac{16y}{3} to -27y.

$$-\frac{97}{3}y+\frac{128}{3}=-22$$

Subtract \frac{128}{3} from both sides of the equation.

$$-\frac{97}{3}y=-\frac{194}{3}$$

Divide both sides of the equation by -\frac{97}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

Substitute 2 for y in x=-\frac{1}{3}y+\frac{8}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{3}\times 2+\frac{8}{3}$$

Multiply -\frac{1}{3} times 2.

$$x=\frac{-2+8}{3}$$

Add \frac{8}{3} to -\frac{2}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=2$$

The system is now solved.

$$x=2,y=2$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 3x-2 by 2y+3.

$$6xy+9x-4y-6=\left(6x-1\right)\left(y+3\right)-27$$

Use the distributive property to multiply 6x-1 by y+3.

$$6xy+9x-4y-6=6xy+18x-y-3-27$$

Subtract 27 from -3 to get -30.

$$6xy+9x-4y-6=6xy+18x-y-30$$

Subtract 6xy from both sides.

$$6xy+9x-4y-6-6xy=18x-y-30$$

Combine 6xy and -6xy to get 0.

$$9x-4y-6=18x-y-30$$

Subtract 18x from both sides.

$$9x-4y-6-18x=-y-30$$

Combine 9x and -18x to get -9x.

$$-9x-4y-6=-y-30$$

Add y to both sides.

$$-9x-4y-6+y=-30$$

Combine -4y and y to get -3y.

$$-9x-3y-6=-30$$

Add 6 to both sides.

$$-9x-3y=-30+6$$

Add -30 and 6 to get -24.

$$-9x-3y=-24$$

Consider the second equation. Use the distributive property to multiply 4x+1 by 3y+2.

$$12xy+8x+3y+2=\left(2x+5\right)\left(6y-4\right)$$

Use the distributive property to multiply 2x+5 by 6y-4.

$$12xy+8x+3y+2=12xy-8x+30y-20$$

Subtract 12xy from both sides.

$$12xy+8x+3y+2-12xy=-8x+30y-20$$

Combine 12xy and -12xy to get 0.

$$8x+3y+2=-8x+30y-20$$

Add 8x to both sides.

$$8x+3y+2+8x=30y-20$$

Combine 8x and 8x to get 16x.

$$16x+3y+2=30y-20$$

Subtract 30y from both sides.

$$16x+3y+2-30y=-20$$

Combine 3y and -30y to get -27y.

$$16x-27y+2=-20$$

Subtract 2 from both sides.

$$16x-27y=-20-2$$

Subtract 2 from -20 to get -22.

$$16x-27y=-22$$

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

$$-9x-3y=-24,16x-27y=-22$$

Write the equations in matrix form.

$$\left(\begin{matrix}-9&-3\\16&-27\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24\\-22\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-9&-3\\16&-27\end{matrix}\right).

$$inverse(\left(\begin{matrix}-9&-3\\16&-27\end{matrix}\right))\left(\begin{matrix}-9&-3\\16&-27\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-3\\16&-27\end{matrix}\right))\left(\begin{matrix}-24\\-22\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-3\\16&-27\end{matrix}\right))\left(\begin{matrix}-24\\-22\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-3\\16&-27\end{matrix}\right))\left(\begin{matrix}-24\\-22\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{27}{-9\left(-27\right)-\left(-3\times 16\right)}&-\frac{-3}{-9\left(-27\right)-\left(-3\times 16\right)}\\-\frac{16}{-9\left(-27\right)-\left(-3\times 16\right)}&-\frac{9}{-9\left(-27\right)-\left(-3\times 16\right)}\end{matrix}\right)\left(\begin{matrix}-24\\-22\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{97}&\frac{1}{97}\\-\frac{16}{291}&-\frac{3}{97}\end{matrix}\right)\left(\begin{matrix}-24\\-22\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{97}\left(-24\right)+\frac{1}{97}\left(-22\right)\\-\frac{16}{291}\left(-24\right)-\frac{3}{97}\left(-22\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\2\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=2,y=2$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 3x-2 by 2y+3.

$$6xy+9x-4y-6=\left(6x-1\right)\left(y+3\right)-27$$

Use the distributive property to multiply 6x-1 by y+3.

$$6xy+9x-4y-6=6xy+18x-y-3-27$$

Subtract 27 from -3 to get -30.

$$6xy+9x-4y-6=6xy+18x-y-30$$

Subtract 6xy from both sides.

$$6xy+9x-4y-6-6xy=18x-y-30$$

Combine 6xy and -6xy to get 0.

$$9x-4y-6=18x-y-30$$

Subtract 18x from both sides.

$$9x-4y-6-18x=-y-30$$

Combine 9x and -18x to get -9x.

$$-9x-4y-6=-y-30$$

Add y to both sides.

$$-9x-4y-6+y=-30$$

Combine -4y and y to get -3y.

$$-9x-3y-6=-30$$

Add 6 to both sides.

$$-9x-3y=-30+6$$

Add -30 and 6 to get -24.

$$-9x-3y=-24$$

Consider the second equation. Use the distributive property to multiply 4x+1 by 3y+2.

$$12xy+8x+3y+2=\left(2x+5\right)\left(6y-4\right)$$

Use the distributive property to multiply 2x+5 by 6y-4.

$$12xy+8x+3y+2=12xy-8x+30y-20$$

Subtract 12xy from both sides.

$$12xy+8x+3y+2-12xy=-8x+30y-20$$

Combine 12xy and -12xy to get 0.

$$8x+3y+2=-8x+30y-20$$

Add 8x to both sides.

$$8x+3y+2+8x=30y-20$$

Combine 8x and 8x to get 16x.

$$16x+3y+2=30y-20$$

Subtract 30y from both sides.

$$16x+3y+2-30y=-20$$

Combine 3y and -30y to get -27y.

$$16x-27y+2=-20$$

Subtract 2 from both sides.

$$16x-27y=-20-2$$

Subtract 2 from -20 to get -22.

$$16x-27y=-22$$

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.

$$-9x-3y=-24,16x-27y=-22$$

To make -9x and 16x equal, multiply all terms on each side of the first equation by 16 and all terms on each side of the second by -9.

$$16\left(-9\right)x+16\left(-3\right)y=16\left(-24\right),-9\times 16x-9\left(-27\right)y=-9\left(-22\right)$$

Simplify.

$$-144x-48y=-384,-144x+243y=198$$

Subtract -144x+243y=198 from -144x-48y=-384 by subtracting like terms on each side of the equal sign.

$$-144x+144x-48y-243y=-384-198$$

Add -144x to 144x. Terms -144x and 144x cancel out, leaving an equation with only one variable that can be solved.

$$-48y-243y=-384-198$$

Add -48y to -243y.

$$-291y=-384-198$$

Add -384 to -198.

$$-291y=-582$$

Divide both sides by -291.

$$y=2$$

Substitute 2 for y in 16x-27y=-22. Because the resulting equation contains only one variable, you can solve for x directly.

$$16x-27\times 2=-22$$

Multiply -27 times 2.

$$16x-54=-22$$

Add 54 to both sides of the equation.

$$16x=32$$

Divide both sides by 16.

$$x=2$$

The system is now solved.

$$x=2,y=2$$