Solve {l}{x+2/5-9-x/3=y-6/4}{x+2-2x+y/8=frac{x+13}{4}}

$$\left\{ \begin{array} { l } { \frac { x + 2 } { 5 } - \frac { 9 - x } { 3 } = \frac { y - 6 } { 4 } } \\ { x + 2 - \frac { 2 x + y } { 8 } = \frac { x + 13 } { 4 } } \end{array} \right.$$

Solve for x, y

$$x=3$$

$$y=2$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 60, the least common multiple of 5,3,4.

$$12\left(x+2\right)-20\left(9-x\right)=15\left(y-6\right)$$

Use the distributive property to multiply 12 by x+2.

$$12x+24-20\left(9-x\right)=15\left(y-6\right)$$

Use the distributive property to multiply -20 by 9-x.

$$12x+24-180+20x=15\left(y-6\right)$$

Subtract 180 from 24 to get -156.

$$12x-156+20x=15\left(y-6\right)$$

Combine 12x and 20x to get 32x.

$$32x-156=15\left(y-6\right)$$

Use the distributive property to multiply 15 by y-6.

$$32x-156=15y-90$$

Subtract 15y from both sides.

$$32x-156-15y=-90$$

Add 156 to both sides.

$$32x-15y=-90+156$$

Add -90 and 156 to get 66.

$$32x-15y=66$$

Consider the second equation. Multiply both sides of the equation by 8, the least common multiple of 8,4.

$$8x+16-\left(2x+y\right)=2\left(x+13\right)$$

To find the opposite of 2x+y, find the opposite of each term.

$$8x+16-2x-y=2\left(x+13\right)$$

Combine 8x and -2x to get 6x.

$$6x+16-y=2\left(x+13\right)$$

Use the distributive property to multiply 2 by x+13.

$$6x+16-y=2x+26$$

Subtract 2x from both sides.

$$6x+16-y-2x=26$$

Combine 6x and -2x to get 4x.

$$4x+16-y=26$$

Subtract 16 from both sides.

$$4x-y=26-16$$

Subtract 16 from 26 to get 10.

$$4x-y=10$$

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.

$$32x-15y=66,4x-y=10$$

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

$$32x-15y=66$$

Add 15y to both sides of the equation.

$$32x=15y+66$$

Divide both sides by 32.

$$x=\frac{1}{32}\left(15y+66\right)$$

Multiply \frac{1}{32} times 15y+66.

$$x=\frac{15}{32}y+\frac{33}{16}$$

Substitute \frac{15y}{32}+\frac{33}{16} for x in the other equation, 4x-y=10.

$$4\left(\frac{15}{32}y+\frac{33}{16}\right)-y=10$$

Multiply 4 times \frac{15y}{32}+\frac{33}{16}.

$$\frac{15}{8}y+\frac{33}{4}-y=10$$

Add \frac{15y}{8} to -y.

$$\frac{7}{8}y+\frac{33}{4}=10$$

Subtract \frac{33}{4} from both sides of the equation.

$$\frac{7}{8}y=\frac{7}{4}$$

Divide both sides of the equation by \frac{7}{8}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

Substitute 2 for y in x=\frac{15}{32}y+\frac{33}{16}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{15}{32}\times 2+\frac{33}{16}$$

Multiply \frac{15}{32} times 2.

$$x=\frac{15+33}{16}$$

Add \frac{33}{16} to \frac{15}{16} 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.

$$x=3,y=2$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 60, the least common multiple of 5,3,4.

$$12\left(x+2\right)-20\left(9-x\right)=15\left(y-6\right)$$

Use the distributive property to multiply 12 by x+2.

$$12x+24-20\left(9-x\right)=15\left(y-6\right)$$

Use the distributive property to multiply -20 by 9-x.

$$12x+24-180+20x=15\left(y-6\right)$$

Subtract 180 from 24 to get -156.

$$12x-156+20x=15\left(y-6\right)$$

Combine 12x and 20x to get 32x.

$$32x-156=15\left(y-6\right)$$

Use the distributive property to multiply 15 by y-6.

$$32x-156=15y-90$$

Subtract 15y from both sides.

$$32x-156-15y=-90$$

Add 156 to both sides.

$$32x-15y=-90+156$$

Add -90 and 156 to get 66.

$$32x-15y=66$$

Consider the second equation. Multiply both sides of the equation by 8, the least common multiple of 8,4.

$$8x+16-\left(2x+y\right)=2\left(x+13\right)$$

To find the opposite of 2x+y, find the opposite of each term.

$$8x+16-2x-y=2\left(x+13\right)$$

Combine 8x and -2x to get 6x.

$$6x+16-y=2\left(x+13\right)$$

Use the distributive property to multiply 2 by x+13.

$$6x+16-y=2x+26$$

Subtract 2x from both sides.

$$6x+16-y-2x=26$$

Combine 6x and -2x to get 4x.

$$4x+16-y=26$$

Subtract 16 from both sides.

$$4x-y=26-16$$

Subtract 16 from 26 to get 10.

$$4x-y=10$$

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

$$32x-15y=66,4x-y=10$$

Write the equations in matrix form.

$$\left(\begin{matrix}32&-15\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}66\\10\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}32&-15\\4&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}32&-15\\4&-1\end{matrix}\right))\left(\begin{matrix}32&-15\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}32&-15\\4&-1\end{matrix}\right))\left(\begin{matrix}66\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}32&-15\\4&-1\end{matrix}\right))\left(\begin{matrix}66\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}32&-15\\4&-1\end{matrix}\right))\left(\begin{matrix}66\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{32\left(-1\right)-\left(-15\times 4\right)}&-\frac{-15}{32\left(-1\right)-\left(-15\times 4\right)}\\-\frac{4}{32\left(-1\right)-\left(-15\times 4\right)}&\frac{32}{32\left(-1\right)-\left(-15\times 4\right)}\end{matrix}\right)\left(\begin{matrix}66\\10\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{28}&\frac{15}{28}\\-\frac{1}{7}&\frac{8}{7}\end{matrix}\right)\left(\begin{matrix}66\\10\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{28}\times 66+\frac{15}{28}\times 10\\-\frac{1}{7}\times 66+\frac{8}{7}\times 10\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\2\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=3,y=2$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 60, the least common multiple of 5,3,4.

$$12\left(x+2\right)-20\left(9-x\right)=15\left(y-6\right)$$

Use the distributive property to multiply 12 by x+2.

$$12x+24-20\left(9-x\right)=15\left(y-6\right)$$

Use the distributive property to multiply -20 by 9-x.

$$12x+24-180+20x=15\left(y-6\right)$$

Subtract 180 from 24 to get -156.

$$12x-156+20x=15\left(y-6\right)$$

Combine 12x and 20x to get 32x.

$$32x-156=15\left(y-6\right)$$

Use the distributive property to multiply 15 by y-6.

$$32x-156=15y-90$$

Subtract 15y from both sides.

$$32x-156-15y=-90$$

Add 156 to both sides.

$$32x-15y=-90+156$$

Add -90 and 156 to get 66.

$$32x-15y=66$$

Consider the second equation. Multiply both sides of the equation by 8, the least common multiple of 8,4.

$$8x+16-\left(2x+y\right)=2\left(x+13\right)$$

To find the opposite of 2x+y, find the opposite of each term.

$$8x+16-2x-y=2\left(x+13\right)$$

Combine 8x and -2x to get 6x.

$$6x+16-y=2\left(x+13\right)$$

Use the distributive property to multiply 2 by x+13.

$$6x+16-y=2x+26$$

Subtract 2x from both sides.

$$6x+16-y-2x=26$$

Combine 6x and -2x to get 4x.

$$4x+16-y=26$$

Subtract 16 from both sides.

$$4x-y=26-16$$

Subtract 16 from 26 to get 10.

$$4x-y=10$$

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.

$$32x-15y=66,4x-y=10$$

To make 32x and 4x equal, multiply all terms on each side of the first equation by 4 and all terms on each side of the second by 32.

$$4\times 32x+4\left(-15\right)y=4\times 66,32\times 4x+32\left(-1\right)y=32\times 10$$

Simplify.

$$128x-60y=264,128x-32y=320$$

Subtract 128x-32y=320 from 128x-60y=264 by subtracting like terms on each side of the equal sign.

$$128x-128x-60y+32y=264-320$$

Add 128x to -128x. Terms 128x and -128x cancel out, leaving an equation with only one variable that can be solved.

$$-60y+32y=264-320$$

Add -60y to 32y.

$$-28y=264-320$$

Add 264 to -320.

$$-28y=-56$$

Divide both sides by -28.

$$y=2$$

Substitute 2 for y in 4x-y=10. Because the resulting equation contains only one variable, you can solve for x directly.

$$4x-2=10$$

Add 2 to both sides of the equation.

$$4x=12$$

Divide both sides by 4.

$$x=3$$

The system is now solved.

$$x=3,y=2$$