Solve {l}{2r+3s=37}{8r+9s=124}

$$\left. \begin{array} { l } { 2 r + 3 s = 37 } \\ { 8 r + 9 s = 124 } \end{array} \right.$$

Solve for r, s

$$r = \frac{13}{2} = 6\frac{1}{2} = 6.5$$

$$s=8$$

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.

$$2r+3s=37,8r+9s=124$$

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

$$2r+3s=37$$

Subtract 3s from both sides of the equation.

$$2r=-3s+37$$

Divide both sides by 2.

$$r=\frac{1}{2}\left(-3s+37\right)$$

Multiply \frac{1}{2} times -3s+37.

$$r=-\frac{3}{2}s+\frac{37}{2}$$

Substitute \frac{-3s+37}{2} for r in the other equation, 8r+9s=124.

$$8\left(-\frac{3}{2}s+\frac{37}{2}\right)+9s=124$$

Multiply 8 times \frac{-3s+37}{2}.

$$-12s+148+9s=124$$

Add -12s to 9s.

$$-3s+148=124$$

Subtract 148 from both sides of the equation.

$$-3s=-24$$

Divide both sides by -3.

$$s=8$$

Substitute 8 for s in r=-\frac{3}{2}s+\frac{37}{2}. Because the resulting equation contains only one variable, you can solve for r directly.

$$r=-\frac{3}{2}\times 8+\frac{37}{2}$$

Multiply -\frac{3}{2} times 8.

$$r=-12+\frac{37}{2}$$

Add \frac{37}{2} to -12.

$$r=\frac{13}{2}$$

The system is now solved.

$$r=\frac{13}{2},s=8$$

Steps Using Matrices

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

$$2r+3s=37,8r+9s=124$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&3\\8&9\end{matrix}\right)\left(\begin{matrix}r\\s\end{matrix}\right)=\left(\begin{matrix}37\\124\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&3\\8&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\8&9\end{matrix}\right))\left(\begin{matrix}2&3\\8&9\end{matrix}\right)\left(\begin{matrix}r\\s\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\8&9\end{matrix}\right))\left(\begin{matrix}37\\124\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}r\\s\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\8&9\end{matrix}\right))\left(\begin{matrix}37\\124\end{matrix}\right)$$

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

$$\left(\begin{matrix}r\\s\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\8&9\end{matrix}\right))\left(\begin{matrix}37\\124\end{matrix}\right)$$

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.

$$\left(\begin{matrix}r\\s\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\times 9-3\times 8}&-\frac{3}{2\times 9-3\times 8}\\-\frac{8}{2\times 9-3\times 8}&\frac{2}{2\times 9-3\times 8}\end{matrix}\right)\left(\begin{matrix}37\\124\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}r\\s\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2}&\frac{1}{2}\\\frac{4}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}37\\124\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}r\\s\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2}\times 37+\frac{1}{2}\times 124\\\frac{4}{3}\times 37-\frac{1}{3}\times 124\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}r\\s\end{matrix}\right)=\left(\begin{matrix}\frac{13}{2}\\8\end{matrix}\right)$$

Extract the matrix elements r and s.

$$r=\frac{13}{2},s=8$$

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.

$$2r+3s=37,8r+9s=124$$

To make 2r and 8r equal, multiply all terms on each side of the first equation by 8 and all terms on each side of the second by 2.

$$8\times 2r+8\times 3s=8\times 37,2\times 8r+2\times 9s=2\times 124$$

Simplify.

$$16r+24s=296,16r+18s=248$$

Subtract 16r+18s=248 from 16r+24s=296 by subtracting like terms on each side of the equal sign.

$$16r-16r+24s-18s=296-248$$

Add 16r to -16r. Terms 16r and -16r cancel out, leaving an equation with only one variable that can be solved.

$$24s-18s=296-248$$

Add 24s to -18s.

$$6s=296-248$$

Add 296 to -248.

$$6s=48$$

Divide both sides by 6.

$$s=8$$

Substitute 8 for s in 8r+9s=124. Because the resulting equation contains only one variable, you can solve for r directly.

$$8r+9\times 8=124$$

Multiply 9 times 8.

$$8r+72=124$$

Subtract 72 from both sides of the equation.

$$8r=52$$

Divide both sides by 8.

$$r=\frac{13}{2}$$

The system is now solved.

$$r=\frac{13}{2},s=8$$