Solve {l}{2x/3-3y/4=1}{y/8-5x/6=2}

$$\left. \begin{array} { l } { \frac { 2 x } { 3 } - \frac { 3 y } { 4 } = 1 } \\ { \frac { y } { 8 } - \frac { 5 x } { 6 } = 2 } \end{array} \right.$$

Solve for x, y

$$x=-3$$

$$y=-4$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4\times 2x-3\times 3y=12$$

Multiply 4 and 2 to get 8.

$$8x-3\times 3y=12$$

Multiply -3 and 3 to get -9.

$$8x-9y=12$$

Consider the second equation. Multiply both sides of the equation by 24, the least common multiple of 8,6.

$$3y-4\times 5x=48$$

Multiply -4 and 5 to get -20.

$$3y-20x=48$$

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.

$$8x-9y=12,-20x+3y=48$$

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

$$8x-9y=12$$

Add 9y to both sides of the equation.

$$8x=9y+12$$

Divide both sides by 8.

$$x=\frac{1}{8}\left(9y+12\right)$$

Multiply \frac{1}{8} times 9y+12.

$$x=\frac{9}{8}y+\frac{3}{2}$$

Substitute \frac{9y}{8}+\frac{3}{2} for x in the other equation, -20x+3y=48.

$$-20\left(\frac{9}{8}y+\frac{3}{2}\right)+3y=48$$

Multiply -20 times \frac{9y}{8}+\frac{3}{2}.

$$-\frac{45}{2}y-30+3y=48$$

Add -\frac{45y}{2} to 3y.

$$-\frac{39}{2}y-30=48$$

Add 30 to both sides of the equation.

$$-\frac{39}{2}y=78$$

Divide both sides of the equation by -\frac{39}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-4$$

Substitute -4 for y in x=\frac{9}{8}y+\frac{3}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{9}{8}\left(-4\right)+\frac{3}{2}$$

Multiply \frac{9}{8} times -4.

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

Add \frac{3}{2} to -\frac{9}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-3$$

The system is now solved.

$$x=-3,y=-4$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4\times 2x-3\times 3y=12$$

Multiply 4 and 2 to get 8.

$$8x-3\times 3y=12$$

Multiply -3 and 3 to get -9.

$$8x-9y=12$$

Consider the second equation. Multiply both sides of the equation by 24, the least common multiple of 8,6.

$$3y-4\times 5x=48$$

Multiply -4 and 5 to get -20.

$$3y-20x=48$$

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

$$8x-9y=12,-20x+3y=48$$

Write the equations in matrix form.

$$\left(\begin{matrix}8&-9\\-20&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\48\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}8&-9\\-20&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-9\\-20&3\end{matrix}\right))\left(\begin{matrix}8&-9\\-20&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-9\\-20&3\end{matrix}\right))\left(\begin{matrix}12\\48\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-9\\-20&3\end{matrix}\right))\left(\begin{matrix}12\\48\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-9\\-20&3\end{matrix}\right))\left(\begin{matrix}12\\48\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8\times 3-\left(-9\left(-20\right)\right)}&-\frac{-9}{8\times 3-\left(-9\left(-20\right)\right)}\\-\frac{-20}{8\times 3-\left(-9\left(-20\right)\right)}&\frac{8}{8\times 3-\left(-9\left(-20\right)\right)}\end{matrix}\right)\left(\begin{matrix}12\\48\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{52}&-\frac{3}{52}\\-\frac{5}{39}&-\frac{2}{39}\end{matrix}\right)\left(\begin{matrix}12\\48\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{52}\times 12-\frac{3}{52}\times 48\\-\frac{5}{39}\times 12-\frac{2}{39}\times 48\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-4\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-3,y=-4$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4\times 2x-3\times 3y=12$$

Multiply 4 and 2 to get 8.

$$8x-3\times 3y=12$$

Multiply -3 and 3 to get -9.

$$8x-9y=12$$

Consider the second equation. Multiply both sides of the equation by 24, the least common multiple of 8,6.

$$3y-4\times 5x=48$$

Multiply -4 and 5 to get -20.

$$3y-20x=48$$

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.

$$8x-9y=12,-20x+3y=48$$

To make 8x and -20x equal, multiply all terms on each side of the first equation by -20 and all terms on each side of the second by 8.

$$-20\times 8x-20\left(-9\right)y=-20\times 12,8\left(-20\right)x+8\times 3y=8\times 48$$

Simplify.

$$-160x+180y=-240,-160x+24y=384$$

Subtract -160x+24y=384 from -160x+180y=-240 by subtracting like terms on each side of the equal sign.

$$-160x+160x+180y-24y=-240-384$$

Add -160x to 160x. Terms -160x and 160x cancel out, leaving an equation with only one variable that can be solved.

$$180y-24y=-240-384$$

Add 180y to -24y.

$$156y=-240-384$$

Add -240 to -384.

$$156y=-624$$

Divide both sides by 156.

$$y=-4$$

Substitute -4 for y in -20x+3y=48. Because the resulting equation contains only one variable, you can solve for x directly.

$$-20x+3\left(-4\right)=48$$

Multiply 3 times -4.

$$-20x-12=48$$

Add 12 to both sides of the equation.

$$-20x=60$$

Divide both sides by -20.

$$x=-3$$

The system is now solved.

$$x=-3,y=-4$$