Solve {l}{a=x(96/80)}{60-a=x+10(96)}

$$\left. \begin{array} { l } { a = x ( 96 / 80 ) } \\ { 60 - a = x + 10 ( 96 ) } \end{array} \right.$$

Solve for a, x

$$x = -\frac{4500}{11} = -409\frac{1}{11} \approx -409.090909091$$

$$a = -\frac{5400}{11} = -490\frac{10}{11} \approx -490.909090909$$

Steps Using Substitution

Consider the first equation. Reduce the fraction \frac{96}{80} to lowest terms by extracting and canceling out 16.

$$a=x\times \frac{6}{5}$$

Subtract x\times \frac{6}{5} from both sides.

$$a-x\times \frac{6}{5}=0$$

Multiply -1 and \frac{6}{5} to get -\frac{6}{5}.

$$a-\frac{6}{5}x=0$$

Consider the second equation. Multiply 10 and 96 to get 960.

$$60-a=x+960$$

Subtract x from both sides.

$$60-a-x=960$$

Subtract 60 from both sides.

$$-a-x=960-60$$

Subtract 60 from 960 to get 900.

$$-a-x=900$$

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.

$$a-\frac{6}{5}x=0,-a-x=900$$

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

$$a-\frac{6}{5}x=0$$

Add \frac{6x}{5} to both sides of the equation.

$$a=\frac{6}{5}x$$

Substitute \frac{6x}{5} for a in the other equation, -a-x=900.

$$-\frac{6}{5}x-x=900$$

Add -\frac{6x}{5} to -x.

$$-\frac{11}{5}x=900$$

Divide both sides of the equation by -\frac{11}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{4500}{11}$$

Substitute -\frac{4500}{11} for x in a=\frac{6}{5}x. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=\frac{6}{5}\left(-\frac{4500}{11}\right)$$

Multiply \frac{6}{5} times -\frac{4500}{11} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$a=-\frac{5400}{11}$$

The system is now solved.

$$a=-\frac{5400}{11},x=-\frac{4500}{11}$$

Steps Using Matrices

Consider the first equation. Reduce the fraction \frac{96}{80} to lowest terms by extracting and canceling out 16.

$$a=x\times \frac{6}{5}$$

Subtract x\times \frac{6}{5} from both sides.

$$a-x\times \frac{6}{5}=0$$

Multiply -1 and \frac{6}{5} to get -\frac{6}{5}.

$$a-\frac{6}{5}x=0$$

Consider the second equation. Multiply 10 and 96 to get 960.

$$60-a=x+960$$

Subtract x from both sides.

$$60-a-x=960$$

Subtract 60 from both sides.

$$-a-x=960-60$$

Subtract 60 from 960 to get 900.

$$-a-x=900$$

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

$$a-\frac{6}{5}x=0,-a-x=900$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-\frac{6}{5}\\-1&-1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}0\\900\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-\frac{6}{5}\\-1&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-\frac{6}{5}\\-1&-1\end{matrix}\right))\left(\begin{matrix}1&-\frac{6}{5}\\-1&-1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{6}{5}\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\900\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{6}{5}\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\900\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{6}{5}\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\900\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-\frac{6}{5}\left(-1\right)\right)}&-\frac{-\frac{6}{5}}{-1-\left(-\frac{6}{5}\left(-1\right)\right)}\\-\frac{-1}{-1-\left(-\frac{6}{5}\left(-1\right)\right)}&\frac{1}{-1-\left(-\frac{6}{5}\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\900\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{11}&-\frac{6}{11}\\-\frac{5}{11}&-\frac{5}{11}\end{matrix}\right)\left(\begin{matrix}0\\900\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{11}\times 900\\-\frac{5}{11}\times 900\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{5400}{11}\\-\frac{4500}{11}\end{matrix}\right)$$

Extract the matrix elements a and x.

$$a=-\frac{5400}{11},x=-\frac{4500}{11}$$

Steps Using Elimination

Consider the first equation. Reduce the fraction \frac{96}{80} to lowest terms by extracting and canceling out 16.

$$a=x\times \frac{6}{5}$$

Subtract x\times \frac{6}{5} from both sides.

$$a-x\times \frac{6}{5}=0$$

Multiply -1 and \frac{6}{5} to get -\frac{6}{5}.

$$a-\frac{6}{5}x=0$$

Consider the second equation. Multiply 10 and 96 to get 960.

$$60-a=x+960$$

Subtract x from both sides.

$$60-a-x=960$$

Subtract 60 from both sides.

$$-a-x=960-60$$

Subtract 60 from 960 to get 900.

$$-a-x=900$$

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.

$$a-\frac{6}{5}x=0,-a-x=900$$

To make a and -a equal, multiply all terms on each side of the first equation by -1 and all terms on each side of the second by 1.

$$-a-\left(-\frac{6}{5}x\right)=0,-a-x=900$$

Simplify.

$$-a+\frac{6}{5}x=0,-a-x=900$$

Subtract -a-x=900 from -a+\frac{6}{5}x=0 by subtracting like terms on each side of the equal sign.

$$-a+a+\frac{6}{5}x+x=-900$$

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

$$\frac{6}{5}x+x=-900$$

Add \frac{6x}{5} to x.

$$\frac{11}{5}x=-900$$

Divide both sides of the equation by \frac{11}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{4500}{11}$$

Substitute -\frac{4500}{11} for x in -a-x=900. Because the resulting equation contains only one variable, you can solve for a directly.

$$-a-\left(-\frac{4500}{11}\right)=900$$

Subtract \frac{4500}{11} from both sides of the equation.

$$-a=\frac{5400}{11}$$

Divide both sides by -1.

$$a=-\frac{5400}{11}$$

The system is now solved.

$$a=-\frac{5400}{11},x=-\frac{4500}{11}$$