Solve {l}{+6y+5z=820}{+3y+2z=610}

$$\left. \begin{array} { l } { + 6 y + 5 z = 820 } \\ { + 3 y + 2 z = 610 } \end{array} \right.$$

Solve for y, z

$$y=470$$

$$z=-400$$

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.

$$6y+5z=820,3y+2z=610$$

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

$$6y+5z=820$$

Subtract 5z from both sides of the equation.

$$6y=-5z+820$$

Divide both sides by 6.

$$y=\frac{1}{6}\left(-5z+820\right)$$

Multiply \frac{1}{6} times -5z+820.

$$y=-\frac{5}{6}z+\frac{410}{3}$$

Substitute -\frac{5z}{6}+\frac{410}{3} for y in the other equation, 3y+2z=610.

$$3\left(-\frac{5}{6}z+\frac{410}{3}\right)+2z=610$$

Multiply 3 times -\frac{5z}{6}+\frac{410}{3}.

$$-\frac{5}{2}z+410+2z=610$$

Add -\frac{5z}{2} to 2z.

$$-\frac{1}{2}z+410=610$$

Subtract 410 from both sides of the equation.

$$-\frac{1}{2}z=200$$

Multiply both sides by -2.

$$z=-400$$

Substitute -400 for z in y=-\frac{5}{6}z+\frac{410}{3}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=-\frac{5}{6}\left(-400\right)+\frac{410}{3}$$

Multiply -\frac{5}{6} times -400.

$$y=\frac{1000+410}{3}$$

Add \frac{410}{3} to \frac{1000}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=470$$

The system is now solved.

$$y=470,z=-400$$

Steps Using Matrices

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

$$6y+5z=820,3y+2z=610$$

Write the equations in matrix form.

$$\left(\begin{matrix}6&5\\3&2\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}820\\610\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}6&5\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&5\\3&2\end{matrix}\right))\left(\begin{matrix}6&5\\3&2\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\3&2\end{matrix}\right))\left(\begin{matrix}820\\610\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\3&2\end{matrix}\right))\left(\begin{matrix}820\\610\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\3&2\end{matrix}\right))\left(\begin{matrix}820\\610\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{2}{6\times 2-5\times 3}&-\frac{5}{6\times 2-5\times 3}\\-\frac{3}{6\times 2-5\times 3}&\frac{6}{6\times 2-5\times 3}\end{matrix}\right)\left(\begin{matrix}820\\610\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}&\frac{5}{3}\\1&-2\end{matrix}\right)\left(\begin{matrix}820\\610\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}\times 820+\frac{5}{3}\times 610\\820-2\times 610\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}470\\-400\end{matrix}\right)$$

Extract the matrix elements y and z.

$$y=470,z=-400$$

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.

$$6y+5z=820,3y+2z=610$$

To make 6y and 3y equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 6.

$$3\times 6y+3\times 5z=3\times 820,6\times 3y+6\times 2z=6\times 610$$

Simplify.

$$18y+15z=2460,18y+12z=3660$$

Subtract 18y+12z=3660 from 18y+15z=2460 by subtracting like terms on each side of the equal sign.

$$18y-18y+15z-12z=2460-3660$$

Add 18y to -18y. Terms 18y and -18y cancel out, leaving an equation with only one variable that can be solved.

$$15z-12z=2460-3660$$

Add 15z to -12z.

$$3z=2460-3660$$

Add 2460 to -3660.

$$3z=-1200$$

Divide both sides by 3.

$$z=-400$$

Substitute -400 for z in 3y+2z=610. Because the resulting equation contains only one variable, you can solve for y directly.

$$3y+2\left(-400\right)=610$$

Multiply 2 times -400.

$$3y-800=610$$

Add 800 to both sides of the equation.

$$3y=1410$$

Divide both sides by 3.

$$y=470$$

The system is now solved.

$$y=470,z=-400$$