Solve {l}{2a+b=420}{11a+5b=1848}

$$\left. \begin{array} { l } { 2 a + b = 420 } \\ { 11 a + 5 b = 1848 } \end{array} \right.$$

Solve for a, b

$$a=-252$$

$$b=924$$

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.

$$2a+b=420,11a+5b=1848$$

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

$$2a+b=420$$

Subtract b from both sides of the equation.

$$2a=-b+420$$

Divide both sides by 2.

$$a=\frac{1}{2}\left(-b+420\right)$$

Multiply \frac{1}{2} times -b+420.

$$a=-\frac{1}{2}b+210$$

Substitute -\frac{b}{2}+210 for a in the other equation, 11a+5b=1848.

$$11\left(-\frac{1}{2}b+210\right)+5b=1848$$

Multiply 11 times -\frac{b}{2}+210.

$$-\frac{11}{2}b+2310+5b=1848$$

Add -\frac{11b}{2} to 5b.

$$-\frac{1}{2}b+2310=1848$$

Subtract 2310 from both sides of the equation.

$$-\frac{1}{2}b=-462$$

Multiply both sides by -2.

$$b=924$$

Substitute 924 for b in a=-\frac{1}{2}b+210. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=-\frac{1}{2}\times 924+210$$

Multiply -\frac{1}{2} times 924.

$$a=-462+210$$

Add 210 to -462.

$$a=-252$$

The system is now solved.

$$a=-252,b=924$$

Steps Using Matrices

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

$$2a+b=420,11a+5b=1848$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&1\\11&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}420\\1848\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&1\\11&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\11&5\end{matrix}\right))\left(\begin{matrix}2&1\\11&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\11&5\end{matrix}\right))\left(\begin{matrix}420\\1848\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\11&5\end{matrix}\right))\left(\begin{matrix}420\\1848\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\11&5\end{matrix}\right))\left(\begin{matrix}420\\1848\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-11}&-\frac{1}{2\times 5-11}\\-\frac{11}{2\times 5-11}&\frac{2}{2\times 5-11}\end{matrix}\right)\left(\begin{matrix}420\\1848\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-5&1\\11&-2\end{matrix}\right)\left(\begin{matrix}420\\1848\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-5\times 420+1848\\11\times 420-2\times 1848\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-252\\924\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=-252,b=924$$

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.

$$2a+b=420,11a+5b=1848$$

To make 2a and 11a equal, multiply all terms on each side of the first equation by 11 and all terms on each side of the second by 2.

$$11\times 2a+11b=11\times 420,2\times 11a+2\times 5b=2\times 1848$$

Simplify.

$$22a+11b=4620,22a+10b=3696$$

Subtract 22a+10b=3696 from 22a+11b=4620 by subtracting like terms on each side of the equal sign.

$$22a-22a+11b-10b=4620-3696$$

Add 22a to -22a. Terms 22a and -22a cancel out, leaving an equation with only one variable that can be solved.

$$11b-10b=4620-3696$$

Add 11b to -10b.

$$b=4620-3696$$

Add 4620 to -3696.

$$b=924$$

Substitute 924 for b in 11a+5b=1848. Because the resulting equation contains only one variable, you can solve for a directly.

$$11a+5\times 924=1848$$

Multiply 5 times 924.

$$11a+4620=1848$$

Subtract 4620 from both sides of the equation.

$$11a=-2772$$

Divide both sides by 11.

$$a=-252$$

The system is now solved.

$$a=-252,b=924$$