Solve {l}{2.50p+3b=358}{p=2b-20}

$$\left. \begin{array} { l } { 2.50 p + 3 b = 358 } \\ { p = 2 b - 20 } \end{array} \right.$$

Solve for p, b

$$p=82$$

$$b=51$$

Steps Using Substitution

Consider the second equation. Subtract 2b from both sides.

$$p-2b=-20$$

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.

$$2.5p+3b=358,p-2b=-20$$

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

$$2.5p+3b=358$$

Subtract 3b from both sides of the equation.

$$2.5p=-3b+358$$

Divide both sides of the equation by 2.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$p=0.4\left(-3b+358\right)$$

Multiply 0.4 times -3b+358.

$$p=-1.2b+143.2$$

Substitute \frac{-6b+716}{5} for p in the other equation, p-2b=-20.

$$-1.2b+143.2-2b=-20$$

Add -\frac{6b}{5} to -2b.

$$-3.2b+143.2=-20$$

Subtract 143.2 from both sides of the equation.

$$-3.2b=-163.2$$

Divide both sides of the equation by -3.2, which is the same as multiplying both sides by the reciprocal of the fraction.

$$b=51$$

Substitute 51 for b in p=-1.2b+143.2. Because the resulting equation contains only one variable, you can solve for p directly.

$$p=-1.2\times 51+143.2$$

Multiply -1.2 times 51.

$$p=\frac{-306+716}{5}$$

Add 143.2 to -61.2 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p=82$$

The system is now solved.

$$p=82,b=51$$

Steps Using Matrices

Consider the second equation. Subtract 2b from both sides.

$$p-2b=-20$$

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

$$2.5p+3b=358,p-2b=-20$$

Write the equations in matrix form.

$$\left(\begin{matrix}2.5&3\\1&-2\end{matrix}\right)\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}358\\-20\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2.5&3\\1&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2.5&3\\1&-2\end{matrix}\right))\left(\begin{matrix}2.5&3\\1&-2\end{matrix}\right)\left(\begin{matrix}p\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&3\\1&-2\end{matrix}\right))\left(\begin{matrix}358\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}p\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&3\\1&-2\end{matrix}\right))\left(\begin{matrix}358\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}p\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&3\\1&-2\end{matrix}\right))\left(\begin{matrix}358\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{2.5\left(-2\right)-3}&-\frac{3}{2.5\left(-2\right)-3}\\-\frac{1}{2.5\left(-2\right)-3}&\frac{2.5}{2.5\left(-2\right)-3}\end{matrix}\right)\left(\begin{matrix}358\\-20\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{3}{8}\\\frac{1}{8}&-\frac{5}{16}\end{matrix}\right)\left(\begin{matrix}358\\-20\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 358+\frac{3}{8}\left(-20\right)\\\frac{1}{8}\times 358-\frac{5}{16}\left(-20\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}p\\b\end{matrix}\right)=\left(\begin{matrix}82\\51\end{matrix}\right)$$

Extract the matrix elements p and b.

$$p=82,b=51$$

Steps Using Elimination

Consider the second equation. Subtract 2b from both sides.

$$p-2b=-20$$

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.

$$2.5p+3b=358,p-2b=-20$$

To make \frac{5p}{2} and p equal, multiply all terms on each side of the first equation by 1 and all terms on each side of the second by 2.5.

$$2.5p+3b=358,2.5p+2.5\left(-2\right)b=2.5\left(-20\right)$$

Simplify.

$$2.5p+3b=358,2.5p-5b=-50$$

Subtract 2.5p-5b=-50 from 2.5p+3b=358 by subtracting like terms on each side of the equal sign.

$$2.5p-2.5p+3b+5b=358+50$$

Add \frac{5p}{2} to -\frac{5p}{2}. Terms \frac{5p}{2} and -\frac{5p}{2} cancel out, leaving an equation with only one variable that can be solved.

$$3b+5b=358+50$$

Add 3b to 5b.

$$8b=358+50$$

Add 358 to 50.

$$8b=408$$

Divide both sides by 8.

$$b=51$$

Substitute 51 for b in p-2b=-20. Because the resulting equation contains only one variable, you can solve for p directly.

$$p-2\times 51=-20$$

Multiply -2 times 51.

$$p-102=-20$$

Add 102 to both sides of the equation.

$$p=82$$

The system is now solved.

$$p=82,b=51$$