Solve {l}{a/2=b/3}{a+b=85}

$$\left. \begin{array} { l } { \frac { a } { 2 } = \frac { b } { 3 } } \\ { a + b = 85 } \end{array} \right.$$

Solve for a, b

$$a=34$$

$$b=51$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3a=2b$$

Divide both sides by 3.

$$a=\frac{1}{3}\times 2b$$

Multiply \frac{1}{3} times 2b.

$$a=\frac{2}{3}b$$

Substitute \frac{2b}{3} for a in the other equation, a+b=85.

$$\frac{2}{3}b+b=85$$

Add \frac{2b}{3} to b.

$$\frac{5}{3}b=85$$

Divide both sides of the equation by \frac{5}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$b=51$$

Substitute 51 for b in a=\frac{2}{3}b. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=\frac{2}{3}\times 51$$

Multiply \frac{2}{3} times 51.

$$a=34$$

The system is now solved.

$$a=34,b=51$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3a=2b$$

Subtract 2b from both sides.

$$3a-2b=0$$

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

$$3a-2b=0,a+b=85$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-2\\1&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}0\\85\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-2\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\1&1\end{matrix}\right))\left(\begin{matrix}3&-2\\1&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&1\end{matrix}\right))\left(\begin{matrix}0\\85\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&1\end{matrix}\right))\left(\begin{matrix}0\\85\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\1&1\end{matrix}\right))\left(\begin{matrix}0\\85\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-2\right)}&-\frac{-2}{3-\left(-2\right)}\\-\frac{1}{3-\left(-2\right)}&\frac{3}{3-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}0\\85\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{2}{5}\\-\frac{1}{5}&\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}0\\85\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 85\\\frac{3}{5}\times 85\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}34\\51\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=34,b=51$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3a=2b$$

Subtract 2b from both sides.

$$3a-2b=0$$

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.

$$3a-2b=0,a+b=85$$

To make 3a 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 3.

$$3a-2b=0,3a+3b=3\times 85$$

Simplify.

$$3a-2b=0,3a+3b=255$$

Subtract 3a+3b=255 from 3a-2b=0 by subtracting like terms on each side of the equal sign.

$$3a-3a-2b-3b=-255$$

Add 3a to -3a. Terms 3a and -3a cancel out, leaving an equation with only one variable that can be solved.

$$-2b-3b=-255$$

Add -2b to -3b.

$$-5b=-255$$

Divide both sides by -5.

$$b=51$$

Substitute 51 for b in a+b=85. Because the resulting equation contains only one variable, you can solve for a directly.

$$a+51=85$$

Subtract 51 from both sides of the equation.

$$a=34$$

The system is now solved.

$$a=34,b=51$$