Solve {l}{3x-y/3+2x+y/2=53/6}{x+y/3+frac{x+y}{5}=frac{24}{5}}

$$\left. \begin{array} { l } { \frac { 3 x - y } { 3 } + \frac { 2 x + y } { 2 } = \frac { 53 } { 6 } } \\ { \frac { x + y } { 3 } + \frac { x + y } { 5 } = \frac { 24 } { 5 } } \end{array} \right.$$

Solve for x, y

$$x=4$$

$$y=5$$

Steps Using Substitution

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

$$2\left(3x-y\right)+3\left(2x+y\right)=53$$

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

$$6x-2y+3\left(2x+y\right)=53$$

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

$$6x-2y+6x+3y=53$$

Combine 6x and 6x to get 12x.

$$12x-2y+3y=53$$

Combine -2y and 3y to get y.

$$12x+y=53$$

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

$$5\left(x+y\right)+3\left(x+y\right)=72$$

Use the distributive property to multiply 5 by x+y.

$$5x+5y+3\left(x+y\right)=72$$

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

$$5x+5y+3x+3y=72$$

Combine 5x and 3x to get 8x.

$$8x+5y+3y=72$$

Combine 5y and 3y to get 8y.

$$8x+8y=72$$

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.

$$12x+y=53,8x+8y=72$$

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

$$12x+y=53$$

Subtract y from both sides of the equation.

$$12x=-y+53$$

Divide both sides by 12.

$$x=\frac{1}{12}\left(-y+53\right)$$

Multiply \frac{1}{12} times -y+53.

$$x=-\frac{1}{12}y+\frac{53}{12}$$

Substitute \frac{-y+53}{12} for x in the other equation, 8x+8y=72.

$$8\left(-\frac{1}{12}y+\frac{53}{12}\right)+8y=72$$

Multiply 8 times \frac{-y+53}{12}.

$$-\frac{2}{3}y+\frac{106}{3}+8y=72$$

Add -\frac{2y}{3} to 8y.

$$\frac{22}{3}y+\frac{106}{3}=72$$

Subtract \frac{106}{3} from both sides of the equation.

$$\frac{22}{3}y=\frac{110}{3}$$

Divide both sides of the equation by \frac{22}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=5$$

Substitute 5 for y in x=-\frac{1}{12}y+\frac{53}{12}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{12}\times 5+\frac{53}{12}$$

Multiply -\frac{1}{12} times 5.

$$x=\frac{-5+53}{12}$$

Add \frac{53}{12} to -\frac{5}{12} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=4$$

The system is now solved.

$$x=4,y=5$$

Steps Using Matrices

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

$$2\left(3x-y\right)+3\left(2x+y\right)=53$$

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

$$6x-2y+3\left(2x+y\right)=53$$

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

$$6x-2y+6x+3y=53$$

Combine 6x and 6x to get 12x.

$$12x-2y+3y=53$$

Combine -2y and 3y to get y.

$$12x+y=53$$

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

$$5\left(x+y\right)+3\left(x+y\right)=72$$

Use the distributive property to multiply 5 by x+y.

$$5x+5y+3\left(x+y\right)=72$$

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

$$5x+5y+3x+3y=72$$

Combine 5x and 3x to get 8x.

$$8x+5y+3y=72$$

Combine 5y and 3y to get 8y.

$$8x+8y=72$$

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

$$12x+y=53,8x+8y=72$$

Write the equations in matrix form.

$$\left(\begin{matrix}12&1\\8&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}53\\72\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}12&1\\8&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}12&1\\8&8\end{matrix}\right))\left(\begin{matrix}12&1\\8&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&1\\8&8\end{matrix}\right))\left(\begin{matrix}53\\72\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&1\\8&8\end{matrix}\right))\left(\begin{matrix}53\\72\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&1\\8&8\end{matrix}\right))\left(\begin{matrix}53\\72\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{12\times 8-8}&-\frac{1}{12\times 8-8}\\-\frac{8}{12\times 8-8}&\frac{12}{12\times 8-8}\end{matrix}\right)\left(\begin{matrix}53\\72\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&-\frac{1}{88}\\-\frac{1}{11}&\frac{3}{22}\end{matrix}\right)\left(\begin{matrix}53\\72\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 53-\frac{1}{88}\times 72\\-\frac{1}{11}\times 53+\frac{3}{22}\times 72\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\5\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=4,y=5$$

Steps Using Elimination

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

$$2\left(3x-y\right)+3\left(2x+y\right)=53$$

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

$$6x-2y+3\left(2x+y\right)=53$$

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

$$6x-2y+6x+3y=53$$

Combine 6x and 6x to get 12x.

$$12x-2y+3y=53$$

Combine -2y and 3y to get y.

$$12x+y=53$$

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

$$5\left(x+y\right)+3\left(x+y\right)=72$$

Use the distributive property to multiply 5 by x+y.

$$5x+5y+3\left(x+y\right)=72$$

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

$$5x+5y+3x+3y=72$$

Combine 5x and 3x to get 8x.

$$8x+5y+3y=72$$

Combine 5y and 3y to get 8y.

$$8x+8y=72$$

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.

$$12x+y=53,8x+8y=72$$

To make 12x and 8x equal, multiply all terms on each side of the first equation by 8 and all terms on each side of the second by 12.

$$8\times 12x+8y=8\times 53,12\times 8x+12\times 8y=12\times 72$$

Simplify.

$$96x+8y=424,96x+96y=864$$

Subtract 96x+96y=864 from 96x+8y=424 by subtracting like terms on each side of the equal sign.

$$96x-96x+8y-96y=424-864$$

Add 96x to -96x. Terms 96x and -96x cancel out, leaving an equation with only one variable that can be solved.

$$8y-96y=424-864$$

Add 8y to -96y.

$$-88y=424-864$$

Add 424 to -864.

$$-88y=-440$$

Divide both sides by -88.

$$y=5$$

Substitute 5 for y in 8x+8y=72. Because the resulting equation contains only one variable, you can solve for x directly.

$$8x+8\times 5=72$$

Multiply 8 times 5.

$$8x+40=72$$

Subtract 40 from both sides of the equation.

$$8x=32$$

Divide both sides by 8.

$$x=4$$

The system is now solved.

$$x=4,y=5$$