Solve {l}{7x+63y=97}{63x+17y=143}

$$\left\{ \begin{array} { l } { 7 x + 63 y = 97 } \\ { 63 x + 17 y = 143 } \end{array} \right.$$

Solve for x, y

$$x = \frac{736}{385} = 1\frac{351}{385} \approx 1.911688312$$

$$y = \frac{73}{55} = 1\frac{18}{55} \approx 1.327272727$$

Steps Using Substitution

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.

$$7x+63y=97,63x+17y=143$$

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

$$7x+63y=97$$

Subtract 63y from both sides of the equation.

$$7x=-63y+97$$

Divide both sides by 7.

$$x=\frac{1}{7}\left(-63y+97\right)$$

Multiply \frac{1}{7} times -63y+97.

$$x=-9y+\frac{97}{7}$$

Substitute -9y+\frac{97}{7} for x in the other equation, 63x+17y=143.

$$63\left(-9y+\frac{97}{7}\right)+17y=143$$

Multiply 63 times -9y+\frac{97}{7}.

$$-567y+873+17y=143$$

Add -567y to 17y.

$$-550y+873=143$$

Subtract 873 from both sides of the equation.

$$-550y=-730$$

Divide both sides by -550.

$$y=\frac{73}{55}$$

Substitute \frac{73}{55} for y in x=-9y+\frac{97}{7}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-9\times \frac{73}{55}+\frac{97}{7}$$

Multiply -9 times \frac{73}{55}.

$$x=-\frac{657}{55}+\frac{97}{7}$$

Add \frac{97}{7} to -\frac{657}{55} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{736}{385}$$

The system is now solved.

$$x=\frac{736}{385},y=\frac{73}{55}$$

Steps Using Matrices

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

$$7x+63y=97,63x+17y=143$$

Write the equations in matrix form.

$$\left(\begin{matrix}7&63\\63&17\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}97\\143\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}7&63\\63&17\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&63\\63&17\end{matrix}\right))\left(\begin{matrix}7&63\\63&17\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&63\\63&17\end{matrix}\right))\left(\begin{matrix}97\\143\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&63\\63&17\end{matrix}\right))\left(\begin{matrix}97\\143\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&63\\63&17\end{matrix}\right))\left(\begin{matrix}97\\143\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{7\times 17-63\times 63}&-\frac{63}{7\times 17-63\times 63}\\-\frac{63}{7\times 17-63\times 63}&\frac{7}{7\times 17-63\times 63}\end{matrix}\right)\left(\begin{matrix}97\\143\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{17}{3850}&\frac{9}{550}\\\frac{9}{550}&-\frac{1}{550}\end{matrix}\right)\left(\begin{matrix}97\\143\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{17}{3850}\times 97+\frac{9}{550}\times 143\\\frac{9}{550}\times 97-\frac{1}{550}\times 143\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{736}{385}\\\frac{73}{55}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{736}{385},y=\frac{73}{55}$$

Steps Using Elimination

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.

$$7x+63y=97,63x+17y=143$$

To make 7x and 63x equal, multiply all terms on each side of the first equation by 63 and all terms on each side of the second by 7.

$$63\times 7x+63\times 63y=63\times 97,7\times 63x+7\times 17y=7\times 143$$

Simplify.

$$441x+3969y=6111,441x+119y=1001$$

Subtract 441x+119y=1001 from 441x+3969y=6111 by subtracting like terms on each side of the equal sign.

$$441x-441x+3969y-119y=6111-1001$$

Add 441x to -441x. Terms 441x and -441x cancel out, leaving an equation with only one variable that can be solved.

$$3969y-119y=6111-1001$$

Add 3969y to -119y.

$$3850y=6111-1001$$

Add 6111 to -1001.

$$3850y=5110$$

Divide both sides by 3850.

$$y=\frac{73}{55}$$

Substitute \frac{73}{55} for y in 63x+17y=143. Because the resulting equation contains only one variable, you can solve for x directly.

$$63x+17\times \frac{73}{55}=143$$

Multiply 17 times \frac{73}{55}.

$$63x+\frac{1241}{55}=143$$

Subtract \frac{1241}{55} from both sides of the equation.

$$63x=\frac{6624}{55}$$

Divide both sides by 63.

$$x=\frac{736}{385}$$

The system is now solved.

$$x=\frac{736}{385},y=\frac{73}{55}$$