Solve {l}{8x+18y=1000}{64x+126y=7280}

$$\left\{ \begin{array} { l } { 8 x + 18 y = 1000 } \\ { 64 x + 126 y = 7280 } \end{array} \right.$$

Solve for x, y

$$x=35$$

$$y=40$$

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.

$$8x+18y=1000,64x+126y=7280$$

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

$$8x+18y=1000$$

Subtract 18y from both sides of the equation.

$$8x=-18y+1000$$

Divide both sides by 8.

$$x=\frac{1}{8}\left(-18y+1000\right)$$

Multiply \frac{1}{8} times -18y+1000.

$$x=-\frac{9}{4}y+125$$

Substitute -\frac{9y}{4}+125 for x in the other equation, 64x+126y=7280.

$$64\left(-\frac{9}{4}y+125\right)+126y=7280$$

Multiply 64 times -\frac{9y}{4}+125.

$$-144y+8000+126y=7280$$

Add -144y to 126y.

$$-18y+8000=7280$$

Subtract 8000 from both sides of the equation.

$$-18y=-720$$

Divide both sides by -18.

$$y=40$$

Substitute 40 for y in x=-\frac{9}{4}y+125. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{9}{4}\times 40+125$$

Multiply -\frac{9}{4} times 40.

$$x=-90+125$$

Add 125 to -90.

$$x=35$$

The system is now solved.

$$x=35,y=40$$

Steps Using Matrices

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

$$8x+18y=1000,64x+126y=7280$$

Write the equations in matrix form.

$$\left(\begin{matrix}8&18\\64&126\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1000\\7280\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}8&18\\64&126\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&18\\64&126\end{matrix}\right))\left(\begin{matrix}8&18\\64&126\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&18\\64&126\end{matrix}\right))\left(\begin{matrix}1000\\7280\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&18\\64&126\end{matrix}\right))\left(\begin{matrix}1000\\7280\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&18\\64&126\end{matrix}\right))\left(\begin{matrix}1000\\7280\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{126}{8\times 126-18\times 64}&-\frac{18}{8\times 126-18\times 64}\\-\frac{64}{8\times 126-18\times 64}&\frac{8}{8\times 126-18\times 64}\end{matrix}\right)\left(\begin{matrix}1000\\7280\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{8}&\frac{1}{8}\\\frac{4}{9}&-\frac{1}{18}\end{matrix}\right)\left(\begin{matrix}1000\\7280\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{8}\times 1000+\frac{1}{8}\times 7280\\\frac{4}{9}\times 1000-\frac{1}{18}\times 7280\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}35\\40\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=35,y=40$$

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.

$$8x+18y=1000,64x+126y=7280$$

To make 8x and 64x equal, multiply all terms on each side of the first equation by 64 and all terms on each side of the second by 8.

$$64\times 8x+64\times 18y=64\times 1000,8\times 64x+8\times 126y=8\times 7280$$

Simplify.

$$512x+1152y=64000,512x+1008y=58240$$

Subtract 512x+1008y=58240 from 512x+1152y=64000 by subtracting like terms on each side of the equal sign.

$$512x-512x+1152y-1008y=64000-58240$$

Add 512x to -512x. Terms 512x and -512x cancel out, leaving an equation with only one variable that can be solved.

$$1152y-1008y=64000-58240$$

Add 1152y to -1008y.

$$144y=64000-58240$$

Add 64000 to -58240.

$$144y=5760$$

Divide both sides by 144.

$$y=40$$

Substitute 40 for y in 64x+126y=7280. Because the resulting equation contains only one variable, you can solve for x directly.

$$64x+126\times 40=7280$$

Multiply 126 times 40.

$$64x+5040=7280$$

Subtract 5040 from both sides of the equation.

$$64x=2240$$

Divide both sides by 64.

$$x=35$$

The system is now solved.

$$x=35,y=40$$