Solve {l}{x-3=17(y-3)}{x+3=5(y+3)}

$$\left. \begin{array} { l } { x - 3 = 17 ( y - 3 ) } \\ { x + 3 = 5 ( y + 3 ) } \end{array} \right.$$

Solve for x, y

$$x=37$$

$$y=5$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 17 by y-3.

$$x-3=17y-51$$

Subtract 17y from both sides.

$$x-3-17y=-51$$

Add 3 to both sides.

$$x-17y=-51+3$$

Add -51 and 3 to get -48.

$$x-17y=-48$$

Consider the second equation. Use the distributive property to multiply 5 by y+3.

$$x+3=5y+15$$

Subtract 5y from both sides.

$$x+3-5y=15$$

Subtract 3 from both sides.

$$x-5y=15-3$$

Subtract 3 from 15 to get 12.

$$x-5y=12$$

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.

$$x-17y=-48,x-5y=12$$

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

$$x-17y=-48$$

Add 17y to both sides of the equation.

$$x=17y-48$$

Substitute 17y-48 for x in the other equation, x-5y=12.

$$17y-48-5y=12$$

Add 17y to -5y.

$$12y-48=12$$

Add 48 to both sides of the equation.

$$12y=60$$

Divide both sides by 12.

$$y=5$$

Substitute 5 for y in x=17y-48. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=17\times 5-48$$

Multiply 17 times 5.

$$x=85-48$$

Add -48 to 85.

$$x=37$$

The system is now solved.

$$x=37,y=5$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 17 by y-3.

$$x-3=17y-51$$

Subtract 17y from both sides.

$$x-3-17y=-51$$

Add 3 to both sides.

$$x-17y=-51+3$$

Add -51 and 3 to get -48.

$$x-17y=-48$$

Consider the second equation. Use the distributive property to multiply 5 by y+3.

$$x+3=5y+15$$

Subtract 5y from both sides.

$$x+3-5y=15$$

Subtract 3 from both sides.

$$x-5y=15-3$$

Subtract 3 from 15 to get 12.

$$x-5y=12$$

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

$$x-17y=-48,x-5y=12$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-17\\1&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-48\\12\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-17\\1&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-17\\1&-5\end{matrix}\right))\left(\begin{matrix}1&-17\\1&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-17\\1&-5\end{matrix}\right))\left(\begin{matrix}-48\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-17\\1&-5\end{matrix}\right))\left(\begin{matrix}-48\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-17\\1&-5\end{matrix}\right))\left(\begin{matrix}-48\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-5-\left(-17\right)}&-\frac{-17}{-5-\left(-17\right)}\\-\frac{1}{-5-\left(-17\right)}&\frac{1}{-5-\left(-17\right)}\end{matrix}\right)\left(\begin{matrix}-48\\12\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{12}&\frac{17}{12}\\-\frac{1}{12}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}-48\\12\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{12}\left(-48\right)+\frac{17}{12}\times 12\\-\frac{1}{12}\left(-48\right)+\frac{1}{12}\times 12\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}37\\5\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=37,y=5$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 17 by y-3.

$$x-3=17y-51$$

Subtract 17y from both sides.

$$x-3-17y=-51$$

Add 3 to both sides.

$$x-17y=-51+3$$

Add -51 and 3 to get -48.

$$x-17y=-48$$

Consider the second equation. Use the distributive property to multiply 5 by y+3.

$$x+3=5y+15$$

Subtract 5y from both sides.

$$x+3-5y=15$$

Subtract 3 from both sides.

$$x-5y=15-3$$

Subtract 3 from 15 to get 12.

$$x-5y=12$$

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.

$$x-17y=-48,x-5y=12$$

Subtract x-5y=12 from x-17y=-48 by subtracting like terms on each side of the equal sign.

$$x-x-17y+5y=-48-12$$

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

$$-17y+5y=-48-12$$

Add -17y to 5y.

$$-12y=-48-12$$

Add -48 to -12.

$$-12y=-60$$

Divide both sides by -12.

$$y=5$$

Substitute 5 for y in x-5y=12. Because the resulting equation contains only one variable, you can solve for x directly.

$$x-5\times 5=12$$

Multiply -5 times 5.

$$x-25=12$$

Add 25 to both sides of the equation.

$$x=37$$

The system is now solved.

$$x=37,y=5$$