Solve {l}{10x+y=8y+2}{10y+x=10x+y+9}

$$\left. \begin{array} { l } { 10 x + y = 8 y + 2 } \\ { 10 y + x = 10 x + y + 9 } \end{array} \right.$$

Solve for x, y

$$x=3$$

$$y=4$$

Steps Using Substitution

Consider the first equation. Subtract 8y from both sides.

$$10x+y-8y=2$$

Combine y and -8y to get -7y.

$$10x-7y=2$$

Consider the second equation. Subtract 10x from both sides.

$$10y+x-10x=y+9$$

Combine x and -10x to get -9x.

$$10y-9x=y+9$$

Subtract y from both sides.

$$10y-9x-y=9$$

Combine 10y and -y to get 9y.

$$9y-9x=9$$

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.

$$10x-7y=2,-9x+9y=9$$

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

$$10x-7y=2$$

Add 7y to both sides of the equation.

$$10x=7y+2$$

Divide both sides by 10.

$$x=\frac{1}{10}\left(7y+2\right)$$

Multiply \frac{1}{10} times 7y+2.

$$x=\frac{7}{10}y+\frac{1}{5}$$

Substitute \frac{7y}{10}+\frac{1}{5} for x in the other equation, -9x+9y=9.

$$-9\left(\frac{7}{10}y+\frac{1}{5}\right)+9y=9$$

Multiply -9 times \frac{7y}{10}+\frac{1}{5}.

$$-\frac{63}{10}y-\frac{9}{5}+9y=9$$

Add -\frac{63y}{10} to 9y.

$$\frac{27}{10}y-\frac{9}{5}=9$$

Add \frac{9}{5} to both sides of the equation.

$$\frac{27}{10}y=\frac{54}{5}$$

Divide both sides of the equation by \frac{27}{10}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=4$$

Substitute 4 for y in x=\frac{7}{10}y+\frac{1}{5}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{7}{10}\times 4+\frac{1}{5}$$

Multiply \frac{7}{10} times 4.

$$x=\frac{14+1}{5}$$

Add \frac{1}{5} to \frac{14}{5} 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. Subtract 8y from both sides.

$$10x+y-8y=2$$

Combine y and -8y to get -7y.

$$10x-7y=2$$

Consider the second equation. Subtract 10x from both sides.

$$10y+x-10x=y+9$$

Combine x and -10x to get -9x.

$$10y-9x=y+9$$

Subtract y from both sides.

$$10y-9x-y=9$$

Combine 10y and -y to get 9y.

$$9y-9x=9$$

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

$$10x-7y=2,-9x+9y=9$$

Write the equations in matrix form.

$$\left(\begin{matrix}10&-7\\-9&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\9\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}10&-7\\-9&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&-7\\-9&9\end{matrix}\right))\left(\begin{matrix}10&-7\\-9&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-7\\-9&9\end{matrix}\right))\left(\begin{matrix}2\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-7\\-9&9\end{matrix}\right))\left(\begin{matrix}2\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-7\\-9&9\end{matrix}\right))\left(\begin{matrix}2\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{10\times 9-\left(-7\left(-9\right)\right)}&-\frac{-7}{10\times 9-\left(-7\left(-9\right)\right)}\\-\frac{-9}{10\times 9-\left(-7\left(-9\right)\right)}&\frac{10}{10\times 9-\left(-7\left(-9\right)\right)}\end{matrix}\right)\left(\begin{matrix}2\\9\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{7}{27}\\\frac{1}{3}&\frac{10}{27}\end{matrix}\right)\left(\begin{matrix}2\\9\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 2+\frac{7}{27}\times 9\\\frac{1}{3}\times 2+\frac{10}{27}\times 9\end{matrix}\right)$$

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. Subtract 8y from both sides.

$$10x+y-8y=2$$

Combine y and -8y to get -7y.

$$10x-7y=2$$

Consider the second equation. Subtract 10x from both sides.

$$10y+x-10x=y+9$$

Combine x and -10x to get -9x.

$$10y-9x=y+9$$

Subtract y from both sides.

$$10y-9x-y=9$$

Combine 10y and -y to get 9y.

$$9y-9x=9$$

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.

$$10x-7y=2,-9x+9y=9$$

To make 10x and -9x equal, multiply all terms on each side of the first equation by -9 and all terms on each side of the second by 10.

$$-9\times 10x-9\left(-7\right)y=-9\times 2,10\left(-9\right)x+10\times 9y=10\times 9$$

Simplify.

$$-90x+63y=-18,-90x+90y=90$$

Subtract -90x+90y=90 from -90x+63y=-18 by subtracting like terms on each side of the equal sign.

$$-90x+90x+63y-90y=-18-90$$

Add -90x to 90x. Terms -90x and 90x cancel out, leaving an equation with only one variable that can be solved.

$$63y-90y=-18-90$$

Add 63y to -90y.

$$-27y=-18-90$$

Add -18 to -90.

$$-27y=-108$$

Divide both sides by -27.

$$y=4$$

Substitute 4 for y in -9x+9y=9. Because the resulting equation contains only one variable, you can solve for x directly.

$$-9x+9\times 4=9$$

Multiply 9 times 4.

$$-9x+36=9$$

Subtract 36 from both sides of the equation.

$$-9x=-27$$

Divide both sides by -9.

$$x=3$$

The system is now solved.

$$x=3,y=4$$