Solve {l}{2x+3y=0.05}{64x+56y=1.2}

$$\left\{ \begin{array} { l } { 2 x + 3 y = 0.05 } \\ { 64 x + 56 y = 1.2 } \end{array} \right.$$

Solve for x, y

$$x=0.01$$

$$y=0.01$$

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.

$$2x+3y=0.05,64x+56y=1.2$$

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

$$2x+3y=0.05$$

Subtract 3y from both sides of the equation.

$$2x=-3y+0.05$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-3y+0.05\right)$$

Multiply \frac{1}{2} times -3y+0.05.

$$x=-\frac{3}{2}y+\frac{1}{40}$$

Substitute -\frac{3y}{2}+\frac{1}{40} for x in the other equation, 64x+56y=1.2.

$$64\left(-\frac{3}{2}y+\frac{1}{40}\right)+56y=1.2$$

Multiply 64 times -\frac{3y}{2}+\frac{1}{40}.

$$-96y+\frac{8}{5}+56y=1.2$$

Add -96y to 56y.

$$-40y+\frac{8}{5}=1.2$$

Subtract \frac{8}{5} from both sides of the equation.

$$-40y=-\frac{2}{5}$$

Divide both sides by -40.

$$y=\frac{1}{100}$$

Substitute \frac{1}{100} for y in x=-\frac{3}{2}y+\frac{1}{40}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{2}\times \frac{1}{100}+\frac{1}{40}$$

Multiply -\frac{3}{2} times \frac{1}{100} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{3}{200}+\frac{1}{40}$$

Add \frac{1}{40} to -\frac{3}{200} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{1}{100}$$

The system is now solved.

$$x=\frac{1}{100},y=\frac{1}{100}$$

Steps Using Matrices

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

$$2x+3y=0.05,64x+56y=1.2$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&3\\64&56\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0.05\\1.2\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&3\\64&56\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\64&56\end{matrix}\right))\left(\begin{matrix}2&3\\64&56\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\64&56\end{matrix}\right))\left(\begin{matrix}0.05\\1.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\64&56\end{matrix}\right))\left(\begin{matrix}0.05\\1.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\64&56\end{matrix}\right))\left(\begin{matrix}0.05\\1.2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{56}{2\times 56-3\times 64}&-\frac{3}{2\times 56-3\times 64}\\-\frac{64}{2\times 56-3\times 64}&\frac{2}{2\times 56-3\times 64}\end{matrix}\right)\left(\begin{matrix}0.05\\1.2\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{10}&\frac{3}{80}\\\frac{4}{5}&-\frac{1}{40}\end{matrix}\right)\left(\begin{matrix}0.05\\1.2\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{10}\times 0.05+\frac{3}{80}\times 1.2\\\frac{4}{5}\times 0.05-\frac{1}{40}\times 1.2\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{100}\\\frac{1}{100}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{1}{100},y=\frac{1}{100}$$

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.

$$2x+3y=0.05,64x+56y=1.2$$

To make 2x 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 2.

$$64\times 2x+64\times 3y=64\times 0.05,2\times 64x+2\times 56y=2\times 1.2$$

Simplify.

$$128x+192y=3.2,128x+112y=2.4$$

Subtract 128x+112y=2.4 from 128x+192y=3.2 by subtracting like terms on each side of the equal sign.

$$128x-128x+192y-112y=\frac{16-12}{5}$$

Add 128x to -128x. Terms 128x and -128x cancel out, leaving an equation with only one variable that can be solved.

$$192y-112y=\frac{16-12}{5}$$

Add 192y to -112y.

$$80y=\frac{16-12}{5}$$

Add 3.2 to -2.4 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$80y=0.8$$

Divide both sides by 80.

$$y=\frac{1}{100}$$

Substitute \frac{1}{100} for y in 64x+56y=1.2. Because the resulting equation contains only one variable, you can solve for x directly.

$$64x+56\times \frac{1}{100}=1.2$$

Multiply 56 times \frac{1}{100}.

$$64x+\frac{14}{25}=1.2$$

Subtract \frac{14}{25} from both sides of the equation.

$$64x=\frac{16}{25}$$

Divide both sides by 64.

$$x=\frac{1}{100}$$

The system is now solved.

$$x=\frac{1}{100},y=\frac{1}{100}$$