Solve {l}{12y+68z=8}{-12y+18z=9}

$$\left. \begin{array} { l } { 12 y + 68 z = 8 } \\ { - 12 y + 18 z = 9 } \end{array} \right.$$

Solve for y, z

$$y=-\frac{39}{86}\approx -0.453488372$$

$$z=\frac{17}{86}\approx 0.197674419$$

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.

$$12y+68z=8,-12y+18z=9$$

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

$$12y+68z=8$$

Subtract 68z from both sides of the equation.

$$12y=-68z+8$$

Divide both sides by 12.

$$y=\frac{1}{12}\left(-68z+8\right)$$

Multiply \frac{1}{12} times -68z+8.

$$y=-\frac{17}{3}z+\frac{2}{3}$$

Substitute \frac{-17z+2}{3} for y in the other equation, -12y+18z=9.

$$-12\left(-\frac{17}{3}z+\frac{2}{3}\right)+18z=9$$

Multiply -12 times \frac{-17z+2}{3}.

$$68z-8+18z=9$$

Add 68z to 18z.

$$86z-8=9$$

Add 8 to both sides of the equation.

$$86z=17$$

Divide both sides by 86.

$$z=\frac{17}{86}$$

Substitute \frac{17}{86} for z in y=-\frac{17}{3}z+\frac{2}{3}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=-\frac{17}{3}\times \frac{17}{86}+\frac{2}{3}$$

Multiply -\frac{17}{3} times \frac{17}{86} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=-\frac{289}{258}+\frac{2}{3}$$

Add \frac{2}{3} to -\frac{289}{258} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=-\frac{39}{86}$$

The system is now solved.

$$y=-\frac{39}{86},z=\frac{17}{86}$$

Steps Using Matrices

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

$$12y+68z=8,-12y+18z=9$$

Write the equations in matrix form.

$$\left(\begin{matrix}12&68\\-12&18\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}8\\9\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}12&68\\-12&18\end{matrix}\right).

$$inverse(\left(\begin{matrix}12&68\\-12&18\end{matrix}\right))\left(\begin{matrix}12&68\\-12&18\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}12&68\\-12&18\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}12&68\\-12&18\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}12&68\\-12&18\end{matrix}\right))\left(\begin{matrix}8\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{18}{12\times 18-68\left(-12\right)}&-\frac{68}{12\times 18-68\left(-12\right)}\\-\frac{-12}{12\times 18-68\left(-12\right)}&\frac{12}{12\times 18-68\left(-12\right)}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{3}{172}&-\frac{17}{258}\\\frac{1}{86}&\frac{1}{86}\end{matrix}\right)\left(\begin{matrix}8\\9\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{3}{172}\times 8-\frac{17}{258}\times 9\\\frac{1}{86}\times 8+\frac{1}{86}\times 9\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{39}{86}\\\frac{17}{86}\end{matrix}\right)$$

Extract the matrix elements y and z.

$$y=-\frac{39}{86},z=\frac{17}{86}$$

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.

$$12y+68z=8,-12y+18z=9$$

To make 12y and -12y equal, multiply all terms on each side of the first equation by -12 and all terms on each side of the second by 12.

$$-12\times 12y-12\times 68z=-12\times 8,12\left(-12\right)y+12\times 18z=12\times 9$$

Simplify.

$$-144y-816z=-96,-144y+216z=108$$

Subtract -144y+216z=108 from -144y-816z=-96 by subtracting like terms on each side of the equal sign.

$$-144y+144y-816z-216z=-96-108$$

Add -144y to 144y. Terms -144y and 144y cancel out, leaving an equation with only one variable that can be solved.

$$-816z-216z=-96-108$$

Add -816z to -216z.

$$-1032z=-96-108$$

Add -96 to -108.

$$-1032z=-204$$

Divide both sides by -1032.

$$z=\frac{17}{86}$$

Substitute \frac{17}{86} for z in -12y+18z=9. Because the resulting equation contains only one variable, you can solve for y directly.

$$-12y+18\times \frac{17}{86}=9$$

Multiply 18 times \frac{17}{86}.

$$-12y+\frac{153}{43}=9$$

Subtract \frac{153}{43} from both sides of the equation.

$$-12y=\frac{234}{43}$$

Divide both sides by -12.

$$y=-\frac{39}{86}$$

The system is now solved.

$$y=-\frac{39}{86},z=\frac{17}{86}$$