Solve {l}{50*2.4x+25*2.4y=660}{(50-20)*2.4x+(25+20)*2.4y=540}

$$\left\{ \begin{array} { l } { 50 \times 2.4 x + 25 \times 2.4 y = 660 } \\ { ( 50 - 20 ) \times 2.4 x + ( 25 + 20 ) \times 2.4 y = 540 } \end{array} \right.$$

Solve for x, y

$$x=4.5$$

$$y=2$$

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.

$$120x+60y=660,72x+108y=540$$

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

$$120x+60y=660$$

Subtract 60y from both sides of the equation.

$$120x=-60y+660$$

Divide both sides by 120.

$$x=\frac{1}{120}\left(-60y+660\right)$$

Multiply \frac{1}{120} times -60y+660.

$$x=-\frac{1}{2}y+\frac{11}{2}$$

Substitute \frac{-y+11}{2} for x in the other equation, 72x+108y=540.

$$72\left(-\frac{1}{2}y+\frac{11}{2}\right)+108y=540$$

Multiply 72 times \frac{-y+11}{2}.

$$-36y+396+108y=540$$

Add -36y to 108y.

$$72y+396=540$$

Subtract 396 from both sides of the equation.

$$72y=144$$

Divide both sides by 72.

$$y=2$$

Substitute 2 for y in x=-\frac{1}{2}y+\frac{11}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{2}\times 2+\frac{11}{2}$$

Multiply -\frac{1}{2} times 2.

$$x=-1+\frac{11}{2}$$

Add \frac{11}{2} to -1.

$$x=\frac{9}{2}$$

The system is now solved.

$$x=\frac{9}{2},y=2$$

Steps Using Matrices

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

$$120x+60y=660,72x+108y=540$$

Write the equations in matrix form.

$$\left(\begin{matrix}120&60\\72&108\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}660\\540\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}120&60\\72&108\end{matrix}\right).

$$inverse(\left(\begin{matrix}120&60\\72&108\end{matrix}\right))\left(\begin{matrix}120&60\\72&108\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}120&60\\72&108\end{matrix}\right))\left(\begin{matrix}660\\540\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}120&60\\72&108\end{matrix}\right))\left(\begin{matrix}660\\540\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}120&60\\72&108\end{matrix}\right))\left(\begin{matrix}660\\540\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{108}{120\times 108-60\times 72}&-\frac{60}{120\times 108-60\times 72}\\-\frac{72}{120\times 108-60\times 72}&\frac{120}{120\times 108-60\times 72}\end{matrix}\right)\left(\begin{matrix}660\\540\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{80}&-\frac{1}{144}\\-\frac{1}{120}&\frac{1}{72}\end{matrix}\right)\left(\begin{matrix}660\\540\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{80}\times 660-\frac{1}{144}\times 540\\-\frac{1}{120}\times 660+\frac{1}{72}\times 540\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2}\\2\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{9}{2},y=2$$

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.

$$120x+60y=660,72x+108y=540$$

To make 120x and 72x equal, multiply all terms on each side of the first equation by 72 and all terms on each side of the second by 120.

$$72\times 120x+72\times 60y=72\times 660,120\times 72x+120\times 108y=120\times 540$$

Simplify.

$$8640x+4320y=47520,8640x+12960y=64800$$

Subtract 8640x+12960y=64800 from 8640x+4320y=47520 by subtracting like terms on each side of the equal sign.

$$8640x-8640x+4320y-12960y=47520-64800$$

Add 8640x to -8640x. Terms 8640x and -8640x cancel out, leaving an equation with only one variable that can be solved.

$$4320y-12960y=47520-64800$$

Add 4320y to -12960y.

$$-8640y=47520-64800$$

Add 47520 to -64800.

$$-8640y=-17280$$

Divide both sides by -8640.

$$y=2$$

Substitute 2 for y in 72x+108y=540. Because the resulting equation contains only one variable, you can solve for x directly.

$$72x+108\times 2=540$$

Multiply 108 times 2.

$$72x+216=540$$

Subtract 216 from both sides of the equation.

$$72x=324$$

Divide both sides by 72.

$$x=\frac{9}{2}$$

The system is now solved.

$$x=\frac{9}{2},y=2$$