Solve {l}{17x=y}{(17+1)x-9=y}

$$\left. \begin{array} { l } { 17 x = y } \\ { ( 17 + 1 ) x - 9 = y } \end{array} \right.$$

Solve for x, y

$$x=9$$

$$y=153$$

Steps Using Substitution

Consider the first equation. Subtract y from both sides.

$$17x-y=0$$

Consider the second equation. Add 17 and 1 to get 18.

$$18x-9=y$$

Subtract y from both sides.

$$18x-9-y=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$18x-y=9$$

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.

$$17x-y=0,18x-y=9$$

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

$$17x-y=0$$

Add y to both sides of the equation.

$$17x=y$$

Divide both sides by 17.

$$x=\frac{1}{17}y$$

Substitute \frac{y}{17} for x in the other equation, 18x-y=9.

$$18\times \frac{1}{17}y-y=9$$

Multiply 18 times \frac{y}{17}.

$$\frac{18}{17}y-y=9$$

Add \frac{18y}{17} to -y.

$$\frac{1}{17}y=9$$

Multiply both sides by 17.

$$y=153$$

Substitute 153 for y in x=\frac{1}{17}y. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{17}\times 153$$

Multiply \frac{1}{17} times 153.

$$x=9$$

The system is now solved.

$$x=9,y=153$$

Steps Using Matrices

Consider the first equation. Subtract y from both sides.

$$17x-y=0$$

Consider the second equation. Add 17 and 1 to get 18.

$$18x-9=y$$

Subtract y from both sides.

$$18x-9-y=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$18x-y=9$$

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

$$17x-y=0,18x-y=9$$

Write the equations in matrix form.

$$\left(\begin{matrix}17&-1\\18&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\9\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}17&-1\\18&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}17&-1\\18&-1\end{matrix}\right))\left(\begin{matrix}17&-1\\18&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}17&-1\\18&-1\end{matrix}\right))\left(\begin{matrix}0\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}17&-1\\18&-1\end{matrix}\right))\left(\begin{matrix}0\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}17&-1\\18&-1\end{matrix}\right))\left(\begin{matrix}0\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{17\left(-1\right)-\left(-18\right)}&-\frac{-1}{17\left(-1\right)-\left(-18\right)}\\-\frac{18}{17\left(-1\right)-\left(-18\right)}&\frac{17}{17\left(-1\right)-\left(-18\right)}\end{matrix}\right)\left(\begin{matrix}0\\9\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&1\\-18&17\end{matrix}\right)\left(\begin{matrix}0\\9\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\17\times 9\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\153\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=9,y=153$$

Steps Using Elimination

Consider the first equation. Subtract y from both sides.

$$17x-y=0$$

Consider the second equation. Add 17 and 1 to get 18.

$$18x-9=y$$

Subtract y from both sides.

$$18x-9-y=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$18x-y=9$$

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.

$$17x-y=0,18x-y=9$$

Subtract 18x-y=9 from 17x-y=0 by subtracting like terms on each side of the equal sign.

$$17x-18x-y+y=-9$$

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

$$17x-18x=-9$$

Add 17x to -18x.

$$-x=-9$$

Divide both sides by -1.

$$x=9$$

Substitute 9 for x in 18x-y=9. Because the resulting equation contains only one variable, you can solve for y directly.

$$18\times 9-y=9$$

Multiply 18 times 9.

$$162-y=9$$

Subtract 162 from both sides of the equation.

$$-y=-153$$

Divide both sides by -1.

$$y=153$$

The system is now solved.

$$x=9,y=153$$