Solve {l}{4x-3y/2-3y-2x/3=y+1}{5x-3y/3-2y-3x/5=x+7}

$$\left. \begin{array} { l } { \frac { 4 x - 3 y } { 2 } - \frac { 3 y - 2 x } { 3 } = y + 1 } \\ { \frac { 5 x - 3 y } { 3 } - \frac { 2 y - 3 x } { 5 } = x + 7 } \end{array} \right.$$

Solve for x, y

$$x=33$$

$$y = \frac{174}{7} = 24\frac{6}{7} \approx 24.857142857$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3\left(4x-3y\right)-2\left(3y-2x\right)=6y+6$$

Use the distributive property to multiply 3 by 4x-3y.

$$12x-9y-2\left(3y-2x\right)=6y+6$$

Use the distributive property to multiply -2 by 3y-2x.

$$12x-9y-6y+4x=6y+6$$

Combine -9y and -6y to get -15y.

$$12x-15y+4x=6y+6$$

Combine 12x and 4x to get 16x.

$$16x-15y=6y+6$$

Subtract 6y from both sides.

$$16x-15y-6y=6$$

Combine -15y and -6y to get -21y.

$$16x-21y=6$$

Consider the second equation. Multiply both sides of the equation by 15, the least common multiple of 3,5.

$$5\left(5x-3y\right)-3\left(2y-3x\right)=15x+105$$

Use the distributive property to multiply 5 by 5x-3y.

$$25x-15y-3\left(2y-3x\right)=15x+105$$

Use the distributive property to multiply -3 by 2y-3x.

$$25x-15y-6y+9x=15x+105$$

Combine -15y and -6y to get -21y.

$$25x-21y+9x=15x+105$$

Combine 25x and 9x to get 34x.

$$34x-21y=15x+105$$

Subtract 15x from both sides.

$$34x-21y-15x=105$$

Combine 34x and -15x to get 19x.

$$19x-21y=105$$

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.

$$16x-21y=6,19x-21y=105$$

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

$$16x-21y=6$$

Add 21y to both sides of the equation.

$$16x=21y+6$$

Divide both sides by 16.

$$x=\frac{1}{16}\left(21y+6\right)$$

Multiply \frac{1}{16} times 21y+6.

$$x=\frac{21}{16}y+\frac{3}{8}$$

Substitute \frac{21y}{16}+\frac{3}{8} for x in the other equation, 19x-21y=105.

$$19\left(\frac{21}{16}y+\frac{3}{8}\right)-21y=105$$

Multiply 19 times \frac{21y}{16}+\frac{3}{8}.

$$\frac{399}{16}y+\frac{57}{8}-21y=105$$

Add \frac{399y}{16} to -21y.

$$\frac{63}{16}y+\frac{57}{8}=105$$

Subtract \frac{57}{8} from both sides of the equation.

$$\frac{63}{16}y=\frac{783}{8}$$

Divide both sides of the equation by \frac{63}{16}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{174}{7}$$

Substitute \frac{174}{7} for y in x=\frac{21}{16}y+\frac{3}{8}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{21}{16}\times \frac{174}{7}+\frac{3}{8}$$

Multiply \frac{21}{16} times \frac{174}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{261+3}{8}$$

Add \frac{3}{8} to \frac{261}{8} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=33$$

The system is now solved.

$$x=33,y=\frac{174}{7}$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3\left(4x-3y\right)-2\left(3y-2x\right)=6y+6$$

Use the distributive property to multiply 3 by 4x-3y.

$$12x-9y-2\left(3y-2x\right)=6y+6$$

Use the distributive property to multiply -2 by 3y-2x.

$$12x-9y-6y+4x=6y+6$$

Combine -9y and -6y to get -15y.

$$12x-15y+4x=6y+6$$

Combine 12x and 4x to get 16x.

$$16x-15y=6y+6$$

Subtract 6y from both sides.

$$16x-15y-6y=6$$

Combine -15y and -6y to get -21y.

$$16x-21y=6$$

Consider the second equation. Multiply both sides of the equation by 15, the least common multiple of 3,5.

$$5\left(5x-3y\right)-3\left(2y-3x\right)=15x+105$$

Use the distributive property to multiply 5 by 5x-3y.

$$25x-15y-3\left(2y-3x\right)=15x+105$$

Use the distributive property to multiply -3 by 2y-3x.

$$25x-15y-6y+9x=15x+105$$

Combine -15y and -6y to get -21y.

$$25x-21y+9x=15x+105$$

Combine 25x and 9x to get 34x.

$$34x-21y=15x+105$$

Subtract 15x from both sides.

$$34x-21y-15x=105$$

Combine 34x and -15x to get 19x.

$$19x-21y=105$$

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

$$16x-21y=6,19x-21y=105$$

Write the equations in matrix form.

$$\left(\begin{matrix}16&-21\\19&-21\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\105\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}16&-21\\19&-21\end{matrix}\right).

$$inverse(\left(\begin{matrix}16&-21\\19&-21\end{matrix}\right))\left(\begin{matrix}16&-21\\19&-21\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-21\\19&-21\end{matrix}\right))\left(\begin{matrix}6\\105\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-21\\19&-21\end{matrix}\right))\left(\begin{matrix}6\\105\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-21\\19&-21\end{matrix}\right))\left(\begin{matrix}6\\105\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{21}{16\left(-21\right)-\left(-21\times 19\right)}&-\frac{-21}{16\left(-21\right)-\left(-21\times 19\right)}\\-\frac{19}{16\left(-21\right)-\left(-21\times 19\right)}&\frac{16}{16\left(-21\right)-\left(-21\times 19\right)}\end{matrix}\right)\left(\begin{matrix}6\\105\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{3}\\-\frac{19}{63}&\frac{16}{63}\end{matrix}\right)\left(\begin{matrix}6\\105\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 6+\frac{1}{3}\times 105\\-\frac{19}{63}\times 6+\frac{16}{63}\times 105\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}33\\\frac{174}{7}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=33,y=\frac{174}{7}$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3\left(4x-3y\right)-2\left(3y-2x\right)=6y+6$$

Use the distributive property to multiply 3 by 4x-3y.

$$12x-9y-2\left(3y-2x\right)=6y+6$$

Use the distributive property to multiply -2 by 3y-2x.

$$12x-9y-6y+4x=6y+6$$

Combine -9y and -6y to get -15y.

$$12x-15y+4x=6y+6$$

Combine 12x and 4x to get 16x.

$$16x-15y=6y+6$$

Subtract 6y from both sides.

$$16x-15y-6y=6$$

Combine -15y and -6y to get -21y.

$$16x-21y=6$$

Consider the second equation. Multiply both sides of the equation by 15, the least common multiple of 3,5.

$$5\left(5x-3y\right)-3\left(2y-3x\right)=15x+105$$

Use the distributive property to multiply 5 by 5x-3y.

$$25x-15y-3\left(2y-3x\right)=15x+105$$

Use the distributive property to multiply -3 by 2y-3x.

$$25x-15y-6y+9x=15x+105$$

Combine -15y and -6y to get -21y.

$$25x-21y+9x=15x+105$$

Combine 25x and 9x to get 34x.

$$34x-21y=15x+105$$

Subtract 15x from both sides.

$$34x-21y-15x=105$$

Combine 34x and -15x to get 19x.

$$19x-21y=105$$

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.

$$16x-21y=6,19x-21y=105$$

Subtract 19x-21y=105 from 16x-21y=6 by subtracting like terms on each side of the equal sign.

$$16x-19x-21y+21y=6-105$$

Add -21y to 21y. Terms -21y and 21y cancel out, leaving an equation with only one variable that can be solved.

$$16x-19x=6-105$$

Add 16x to -19x.

$$-3x=6-105$$

Add 6 to -105.

$$-3x=-99$$

Divide both sides by -3.

$$x=33$$

Substitute 33 for x in 19x-21y=105. Because the resulting equation contains only one variable, you can solve for y directly.

$$19\times 33-21y=105$$

Multiply 19 times 33.

$$627-21y=105$$

Subtract 627 from both sides of the equation.

$$-21y=-522$$

Divide both sides by -21.

$$y=\frac{174}{7}$$

The system is now solved.

$$x=33,y=\frac{174}{7}$$