Solve {l}{y/8+x/7=1}{x+2/2-y-4/4=11}

$$\left\{ \begin{array} { l } { \frac { y } { 8 } + \frac { x } { 7 } = 1 } \\ { \frac { x + 2 } { 2 } - \frac { y - 4 } { 4 } = 11 } \end{array} \right.$$

Solve for y, x

$$x=14$$

$$y=-8$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 56, the least common multiple of 8,7.

$$7y+8x=56$$

Consider the second equation. Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(x+2\right)-\left(y-4\right)=44$$

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

$$2x+4-\left(y-4\right)=44$$

To find the opposite of y-4, find the opposite of each term.

$$2x+4-y+4=44$$

Add 4 and 4 to get 8.

$$2x+8-y=44$$

Subtract 8 from both sides.

$$2x-y=44-8$$

Subtract 8 from 44 to get 36.

$$2x-y=36$$

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.

$$7y+8x=56,-y+2x=36$$

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

$$7y+8x=56$$

Subtract 8x from both sides of the equation.

$$7y=-8x+56$$

Divide both sides by 7.

$$y=\frac{1}{7}\left(-8x+56\right)$$

Multiply \frac{1}{7} times -8x+56.

$$y=-\frac{8}{7}x+8$$

Substitute -\frac{8x}{7}+8 for y in the other equation, -y+2x=36.

$$-\left(-\frac{8}{7}x+8\right)+2x=36$$

Multiply -1 times -\frac{8x}{7}+8.

$$\frac{8}{7}x-8+2x=36$$

Add \frac{8x}{7} to 2x.

$$\frac{22}{7}x-8=36$$

Add 8 to both sides of the equation.

$$\frac{22}{7}x=44$$

Divide both sides of the equation by \frac{22}{7}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=14$$

Substitute 14 for x in y=-\frac{8}{7}x+8. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=-\frac{8}{7}\times 14+8$$

Multiply -\frac{8}{7} times 14.

$$y=-16+8$$

Add 8 to -16.

$$y=-8$$

The system is now solved.

$$y=-8,x=14$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 56, the least common multiple of 8,7.

$$7y+8x=56$$

Consider the second equation. Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(x+2\right)-\left(y-4\right)=44$$

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

$$2x+4-\left(y-4\right)=44$$

To find the opposite of y-4, find the opposite of each term.

$$2x+4-y+4=44$$

Add 4 and 4 to get 8.

$$2x+8-y=44$$

Subtract 8 from both sides.

$$2x-y=44-8$$

Subtract 8 from 44 to get 36.

$$2x-y=36$$

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

$$7y+8x=56,-y+2x=36$$

Write the equations in matrix form.

$$\left(\begin{matrix}7&8\\-1&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}56\\36\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}7&8\\-1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&8\\-1&2\end{matrix}\right))\left(\begin{matrix}7&8\\-1&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\-1&2\end{matrix}\right))\left(\begin{matrix}56\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\-1&2\end{matrix}\right))\left(\begin{matrix}56\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\-1&2\end{matrix}\right))\left(\begin{matrix}56\\36\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7\times 2-8\left(-1\right)}&-\frac{8}{7\times 2-8\left(-1\right)}\\-\frac{-1}{7\times 2-8\left(-1\right)}&\frac{7}{7\times 2-8\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}56\\36\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&-\frac{4}{11}\\\frac{1}{22}&\frac{7}{22}\end{matrix}\right)\left(\begin{matrix}56\\36\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 56-\frac{4}{11}\times 36\\\frac{1}{22}\times 56+\frac{7}{22}\times 36\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-8\\14\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=-8,x=14$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 56, the least common multiple of 8,7.

$$7y+8x=56$$

Consider the second equation. Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(x+2\right)-\left(y-4\right)=44$$

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

$$2x+4-\left(y-4\right)=44$$

To find the opposite of y-4, find the opposite of each term.

$$2x+4-y+4=44$$

Add 4 and 4 to get 8.

$$2x+8-y=44$$

Subtract 8 from both sides.

$$2x-y=44-8$$

Subtract 8 from 44 to get 36.

$$2x-y=36$$

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.

$$7y+8x=56,-y+2x=36$$

To make 7y and -y equal, multiply all terms on each side of the first equation by -1 and all terms on each side of the second by 7.

$$-7y-8x=-56,7\left(-1\right)y+7\times 2x=7\times 36$$

Simplify.

$$-7y-8x=-56,-7y+14x=252$$

Subtract -7y+14x=252 from -7y-8x=-56 by subtracting like terms on each side of the equal sign.

$$-7y+7y-8x-14x=-56-252$$

Add -7y to 7y. Terms -7y and 7y cancel out, leaving an equation with only one variable that can be solved.

$$-8x-14x=-56-252$$

Add -8x to -14x.

$$-22x=-56-252$$

Add -56 to -252.

$$-22x=-308$$

Divide both sides by -22.

$$x=14$$

Substitute 14 for x in -y+2x=36. Because the resulting equation contains only one variable, you can solve for y directly.

$$-y+2\times 14=36$$

Multiply 2 times 14.

$$-y+28=36$$

Subtract 28 from both sides of the equation.

$$-y=8$$

Divide both sides by -1.

$$y=-8$$

The system is now solved.

$$y=-8,x=14$$