Solve {l}{a=x(96/6)}{160-a=x+10(96/60)}

$$\left. \begin{array} { l } { a = x ( 96 / 6 ) } \\ { 160 - a = x + 10 ( 96 / 60 ) } \end{array} \right.$$

Solve for a, x

$$x = \frac{160}{17} = 9\frac{7}{17} \approx 9.411764706$$

$$a = \frac{2560}{17} = 150\frac{10}{17} \approx 150.588235294$$

Steps Using Substitution

Consider the first equation. Divide 96 by 6 to get 16.

$$a=x\times 16$$

Subtract x\times 16 from both sides.

$$a-x\times 16=0$$

Multiply -1 and 16 to get -16.

$$a-16x=0$$

Consider the second equation. Divide 96 by 6 to get 16.

$$160-a=x+10\times 16\times 0$$

Multiply 10 and 16 to get 160.

$$160-a=x+160\times 0$$

Multiply 160 and 0 to get 0.

$$160-a=x+0$$

Anything plus zero gives itself.

$$160-a=x$$

Subtract x from both sides.

$$160-a-x=0$$

Subtract 160 from both sides. Anything subtracted from zero gives its negation.

$$-a-x=-160$$

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.

$$a-16x=0,-a-x=-160$$

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

$$a-16x=0$$

Add 16x to both sides of the equation.

$$a=16x$$

Substitute 16x for a in the other equation, -a-x=-160.

$$-16x-x=-160$$

Add -16x to -x.

$$-17x=-160$$

Divide both sides by -17.

$$x=\frac{160}{17}$$

Substitute \frac{160}{17} for x in a=16x. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=16\times \frac{160}{17}$$

Multiply 16 times \frac{160}{17}.

$$a=\frac{2560}{17}$$

The system is now solved.

$$a=\frac{2560}{17},x=\frac{160}{17}$$

Steps Using Matrices

Consider the first equation. Divide 96 by 6 to get 16.

$$a=x\times 16$$

Subtract x\times 16 from both sides.

$$a-x\times 16=0$$

Multiply -1 and 16 to get -16.

$$a-16x=0$$

Consider the second equation. Divide 96 by 6 to get 16.

$$160-a=x+10\times 16\times 0$$

Multiply 10 and 16 to get 160.

$$160-a=x+160\times 0$$

Multiply 160 and 0 to get 0.

$$160-a=x+0$$

Anything plus zero gives itself.

$$160-a=x$$

Subtract x from both sides.

$$160-a-x=0$$

Subtract 160 from both sides. Anything subtracted from zero gives its negation.

$$-a-x=-160$$

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

$$a-16x=0,-a-x=-160$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-16\\-1&-1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}0\\-160\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-16\\-1&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-16\\-1&-1\end{matrix}\right))\left(\begin{matrix}1&-16\\-1&-1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-16\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\-160\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-16\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\-160\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-16\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\-160\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-16\left(-1\right)\right)}&-\frac{-16}{-1-\left(-16\left(-1\right)\right)}\\-\frac{-1}{-1-\left(-16\left(-1\right)\right)}&\frac{1}{-1-\left(-16\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\-160\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}&-\frac{16}{17}\\-\frac{1}{17}&-\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}0\\-160\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{17}\left(-160\right)\\-\frac{1}{17}\left(-160\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2560}{17}\\\frac{160}{17}\end{matrix}\right)$$

Extract the matrix elements a and x.

$$a=\frac{2560}{17},x=\frac{160}{17}$$

Steps Using Elimination

Consider the first equation. Divide 96 by 6 to get 16.

$$a=x\times 16$$

Subtract x\times 16 from both sides.

$$a-x\times 16=0$$

Multiply -1 and 16 to get -16.

$$a-16x=0$$

Consider the second equation. Divide 96 by 6 to get 16.

$$160-a=x+10\times 16\times 0$$

Multiply 10 and 16 to get 160.

$$160-a=x+160\times 0$$

Multiply 160 and 0 to get 0.

$$160-a=x+0$$

Anything plus zero gives itself.

$$160-a=x$$

Subtract x from both sides.

$$160-a-x=0$$

Subtract 160 from both sides. Anything subtracted from zero gives its negation.

$$-a-x=-160$$

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.

$$a-16x=0,-a-x=-160$$

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.

$$-a-\left(-16x\right)=0,-a-x=-160$$

Simplify.

$$-a+16x=0,-a-x=-160$$

Subtract -a-x=-160 from -a+16x=0 by subtracting like terms on each side of the equal sign.

$$-a+a+16x+x=160$$

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

$$16x+x=160$$

Add 16x to x.

$$17x=160$$

Divide both sides by 17.

$$x=\frac{160}{17}$$

Substitute \frac{160}{17} for x in -a-x=-160. Because the resulting equation contains only one variable, you can solve for a directly.

$$-a-\frac{160}{17}=-160$$

Add \frac{160}{17} to both sides of the equation.

$$-a=-\frac{2560}{17}$$

Divide both sides by -1.

$$a=\frac{2560}{17}$$

The system is now solved.

$$a=\frac{2560}{17},x=\frac{160}{17}$$