Solve {l}{64x+144y=m}{(y+x)*64=a}

$$\left. \begin{array} { l } { 64 x + 144 y = m } \\ { ( y + x ) \times 64 = a } \end{array} \right.$$

Solve for x, y

$$x=\frac{9a}{320}-\frac{m}{80}$$

$$y=\frac{m-a}{80}$$

Steps Using Substitution

Consider the second equation. Use the distributive property to multiply y+x by 64.

$$64y+64x=a$$

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.

$$64x+144y=m,64x+64y=a$$

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

$$64x+144y=m$$

Subtract 144y from both sides of the equation.

$$64x=-144y+m$$

Divide both sides by 64.

$$x=\frac{1}{64}\left(-144y+m\right)$$

Multiply \frac{1}{64} times -144y+m.

$$x=-\frac{9}{4}y+\frac{m}{64}$$

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

$$64\left(-\frac{9}{4}y+\frac{m}{64}\right)+64y=a$$

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

$$-144y+m+64y=a$$

Add -144y to 64y.

$$-80y+m=a$$

Subtract m from both sides of the equation.

$$-80y=a-m$$

Divide both sides by -80.

$$y=\frac{m-a}{80}$$

Substitute \frac{-a+m}{80} for y in x=-\frac{9}{4}y+\frac{m}{64}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{9}{4}\times \frac{m-a}{80}+\frac{m}{64}$$

Multiply -\frac{9}{4} times \frac{-a+m}{80}.

$$x=\frac{9a-9m}{320}+\frac{m}{64}$$

Add \frac{m}{64} to \frac{9a-9m}{320}.

$$x=\frac{9a}{320}-\frac{m}{80}$$

The system is now solved.

$$x=\frac{9a}{320}-\frac{m}{80},y=\frac{m-a}{80}$$

Steps Using Matrices

Consider the second equation. Use the distributive property to multiply y+x by 64.

$$64y+64x=a$$

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

$$64x+144y=m,64x+64y=a$$

Write the equations in matrix form.

$$\left(\begin{matrix}64&144\\64&64\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}m\\a\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}64&144\\64&64\end{matrix}\right).

$$inverse(\left(\begin{matrix}64&144\\64&64\end{matrix}\right))\left(\begin{matrix}64&144\\64&64\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}64&144\\64&64\end{matrix}\right))\left(\begin{matrix}m\\a\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}64&144\\64&64\end{matrix}\right))\left(\begin{matrix}m\\a\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}64&144\\64&64\end{matrix}\right))\left(\begin{matrix}m\\a\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{64}{64\times 64-144\times 64}&-\frac{144}{64\times 64-144\times 64}\\-\frac{64}{64\times 64-144\times 64}&\frac{64}{64\times 64-144\times 64}\end{matrix}\right)\left(\begin{matrix}m\\a\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{80}&\frac{9}{320}\\\frac{1}{80}&-\frac{1}{80}\end{matrix}\right)\left(\begin{matrix}m\\a\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{80}m+\frac{9}{320}a\\\frac{1}{80}m-\frac{1}{80}a\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9a}{320}-\frac{m}{80}\\\frac{m-a}{80}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{9a}{320}-\frac{m}{80},y=\frac{m-a}{80}$$

Steps Using Elimination

Consider the second equation. Use the distributive property to multiply y+x by 64.

$$64y+64x=a$$

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.

$$64x+144y=m,64x+64y=a$$

Subtract 64x+64y=a from 64x+144y=m by subtracting like terms on each side of the equal sign.

$$64x-64x+144y-64y=m-a$$

Add 64x to -64x. Terms 64x and -64x cancel out, leaving an equation with only one variable that can be solved.

$$144y-64y=m-a$$

Add 144y to -64y.

$$80y=m-a$$

Divide both sides by 80.

$$y=\frac{m-a}{80}$$

Substitute \frac{m-a}{80} for y in 64x+64y=a. Because the resulting equation contains only one variable, you can solve for x directly.

$$64x+64\times \frac{m-a}{80}=a$$

Multiply 64 times \frac{m-a}{80}.

$$64x+\frac{4m-4a}{5}=a$$

Subtract \frac{4m-4a}{5} from both sides of the equation.

$$64x=\frac{9a-4m}{5}$$

Divide both sides by 64.

$$x=\frac{9a}{320}-\frac{m}{80}$$

The system is now solved.

$$x=\frac{9a}{320}-\frac{m}{80},y=\frac{m-a}{80}$$