Solve {l}{11a=5b+48}{7a=13b-840}

$$\left. \begin{array} { l } { 11 a = 5 b + 48 } \\ { 7 a = 13 b - 840 } \end{array} \right.$$

Solve for a, b

$$a = \frac{134}{3} = 44\frac{2}{3} \approx 44.666666667$$

$$b = \frac{266}{3} = 88\frac{2}{3} \approx 88.666666667$$

Steps Using Substitution

Consider the first equation. Subtract 5b from both sides.

$$11a-5b=48$$

Consider the second equation. Subtract 13b from both sides.

$$7a-13b=-840$$

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.

$$11a-5b=48,7a-13b=-840$$

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

$$11a-5b=48$$

Add 5b to both sides of the equation.

$$11a=5b+48$$

Divide both sides by 11.

$$a=\frac{1}{11}\left(5b+48\right)$$

Multiply \frac{1}{11} times 5b+48.

$$a=\frac{5}{11}b+\frac{48}{11}$$

Substitute \frac{5b+48}{11} for a in the other equation, 7a-13b=-840.

$$7\left(\frac{5}{11}b+\frac{48}{11}\right)-13b=-840$$

Multiply 7 times \frac{5b+48}{11}.

$$\frac{35}{11}b+\frac{336}{11}-13b=-840$$

Add \frac{35b}{11} to -13b.

$$-\frac{108}{11}b+\frac{336}{11}=-840$$

Subtract \frac{336}{11} from both sides of the equation.

$$-\frac{108}{11}b=-\frac{9576}{11}$$

Divide both sides of the equation by -\frac{108}{11}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$b=\frac{266}{3}$$

Substitute \frac{266}{3} for b in a=\frac{5}{11}b+\frac{48}{11}. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=\frac{5}{11}\times \frac{266}{3}+\frac{48}{11}$$

Multiply \frac{5}{11} times \frac{266}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$a=\frac{1330}{33}+\frac{48}{11}$$

Add \frac{48}{11} to \frac{1330}{33} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a=\frac{134}{3}$$

The system is now solved.

$$a=\frac{134}{3},b=\frac{266}{3}$$

Steps Using Matrices

Consider the first equation. Subtract 5b from both sides.

$$11a-5b=48$$

Consider the second equation. Subtract 13b from both sides.

$$7a-13b=-840$$

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

$$11a-5b=48,7a-13b=-840$$

Write the equations in matrix form.

$$\left(\begin{matrix}11&-5\\7&-13\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}48\\-840\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}11&-5\\7&-13\end{matrix}\right).

$$inverse(\left(\begin{matrix}11&-5\\7&-13\end{matrix}\right))\left(\begin{matrix}11&-5\\7&-13\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}11&-5\\7&-13\end{matrix}\right))\left(\begin{matrix}48\\-840\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}11&-5\\7&-13\end{matrix}\right))\left(\begin{matrix}48\\-840\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}11&-5\\7&-13\end{matrix}\right))\left(\begin{matrix}48\\-840\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{11\left(-13\right)-\left(-5\times 7\right)}&-\frac{-5}{11\left(-13\right)-\left(-5\times 7\right)}\\-\frac{7}{11\left(-13\right)-\left(-5\times 7\right)}&\frac{11}{11\left(-13\right)-\left(-5\times 7\right)}\end{matrix}\right)\left(\begin{matrix}48\\-840\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{13}{108}&-\frac{5}{108}\\\frac{7}{108}&-\frac{11}{108}\end{matrix}\right)\left(\begin{matrix}48\\-840\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{13}{108}\times 48-\frac{5}{108}\left(-840\right)\\\frac{7}{108}\times 48-\frac{11}{108}\left(-840\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{134}{3}\\\frac{266}{3}\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=\frac{134}{3},b=\frac{266}{3}$$

Steps Using Elimination

Consider the first equation. Subtract 5b from both sides.

$$11a-5b=48$$

Consider the second equation. Subtract 13b from both sides.

$$7a-13b=-840$$

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.

$$11a-5b=48,7a-13b=-840$$

To make 11a and 7a equal, multiply all terms on each side of the first equation by 7 and all terms on each side of the second by 11.

$$7\times 11a+7\left(-5\right)b=7\times 48,11\times 7a+11\left(-13\right)b=11\left(-840\right)$$

Simplify.

$$77a-35b=336,77a-143b=-9240$$

Subtract 77a-143b=-9240 from 77a-35b=336 by subtracting like terms on each side of the equal sign.

$$77a-77a-35b+143b=336+9240$$

Add 77a to -77a. Terms 77a and -77a cancel out, leaving an equation with only one variable that can be solved.

$$-35b+143b=336+9240$$

Add -35b to 143b.

$$108b=336+9240$$

Add 336 to 9240.

$$108b=9576$$

Divide both sides by 108.

$$b=\frac{266}{3}$$

Substitute \frac{266}{3} for b in 7a-13b=-840. Because the resulting equation contains only one variable, you can solve for a directly.

$$7a-13\times \frac{266}{3}=-840$$

Multiply -13 times \frac{266}{3}.

$$7a-\frac{3458}{3}=-840$$

Add \frac{3458}{3} to both sides of the equation.

$$7a=\frac{938}{3}$$

Divide both sides by 7.

$$a=\frac{134}{3}$$

The system is now solved.

$$a=\frac{134}{3},b=\frac{266}{3}$$