Solve {l}{5a_{0}+5327a_{2}=2714}{157a_{0}+(9233)a_{2}=97761}

$$\left\{ \begin{array} { l } { 5 a _ { 0 } + 5327 a _ { 2 } = 2714 } \\ { 157 a _ { 0 } + ( 9233 ) a _ { 2 } = 97761 } \end{array} \right.$$

Solve for a_0, a_2

$$a_{0} = \frac{70816355}{112882} = 627\frac{39341}{112882} \approx 627.348514378$$

$$a_{2}=-\frac{62707}{790174}\approx -0.07935847$$

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.

$$5a_{0}+5327a_{2}=2714,157a_{0}+9233a_{2}=97761$$

Choose one of the equations and solve it for a_{0} by isolating a_{0} on the left hand side of the equal sign.

$$5a_{0}+5327a_{2}=2714$$

Subtract 5327a_{2} from both sides of the equation.

$$5a_{0}=-5327a_{2}+2714$$

Divide both sides by 5.

$$a_{0}=\frac{1}{5}\left(-5327a_{2}+2714\right)$$

Multiply \frac{1}{5} times -5327a_{2}+2714.

$$a_{0}=-\frac{5327}{5}a_{2}+\frac{2714}{5}$$

Substitute \frac{-5327a_{2}+2714}{5} for a_{0} in the other equation, 157a_{0}+9233a_{2}=97761.

$$157\left(-\frac{5327}{5}a_{2}+\frac{2714}{5}\right)+9233a_{2}=97761$$

Multiply 157 times \frac{-5327a_{2}+2714}{5}.

$$-\frac{836339}{5}a_{2}+\frac{426098}{5}+9233a_{2}=97761$$

Add -\frac{836339a_{2}}{5} to 9233a_{2}.

$$-\frac{790174}{5}a_{2}+\frac{426098}{5}=97761$$

Subtract \frac{426098}{5} from both sides of the equation.

$$-\frac{790174}{5}a_{2}=\frac{62707}{5}$$

Divide both sides of the equation by -\frac{790174}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$a_{2}=-\frac{62707}{790174}$$

Substitute -\frac{62707}{790174} for a_{2} in a_{0}=-\frac{5327}{5}a_{2}+\frac{2714}{5}. Because the resulting equation contains only one variable, you can solve for a_{0} directly.

$$a_{0}=-\frac{5327}{5}\left(-\frac{62707}{790174}\right)+\frac{2714}{5}$$

Multiply -\frac{5327}{5} times -\frac{62707}{790174} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$a_{0}=\frac{47720027}{564410}+\frac{2714}{5}$$

Add \frac{2714}{5} to \frac{47720027}{564410} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a_{0}=\frac{70816355}{112882}$$

The system is now solved.

$$a_{0}=\frac{70816355}{112882},a_{2}=-\frac{62707}{790174}$$

Steps Using Matrices

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

$$5a_{0}+5327a_{2}=2714,157a_{0}+9233a_{2}=97761$$

Write the equations in matrix form.

$$\left(\begin{matrix}5&5327\\157&9233\end{matrix}\right)\left(\begin{matrix}a_{0}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}2714\\97761\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}5&5327\\157&9233\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&5327\\157&9233\end{matrix}\right))\left(\begin{matrix}5&5327\\157&9233\end{matrix}\right)\left(\begin{matrix}a_{0}\\a_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}5&5327\\157&9233\end{matrix}\right))\left(\begin{matrix}2714\\97761\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a_{0}\\a_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}5&5327\\157&9233\end{matrix}\right))\left(\begin{matrix}2714\\97761\end{matrix}\right)$$

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

$$\left(\begin{matrix}a_{0}\\a_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}5&5327\\157&9233\end{matrix}\right))\left(\begin{matrix}2714\\97761\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a_{0}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{9233}{5\times 9233-5327\times 157}&-\frac{5327}{5\times 9233-5327\times 157}\\-\frac{157}{5\times 9233-5327\times 157}&\frac{5}{5\times 9233-5327\times 157}\end{matrix}\right)\left(\begin{matrix}2714\\97761\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a_{0}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}-\frac{1319}{112882}&\frac{761}{112882}\\\frac{157}{790174}&-\frac{5}{790174}\end{matrix}\right)\left(\begin{matrix}2714\\97761\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a_{0}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}-\frac{1319}{112882}\times 2714+\frac{761}{112882}\times 97761\\\frac{157}{790174}\times 2714-\frac{5}{790174}\times 97761\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a_{0}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{70816355}{112882}\\-\frac{62707}{790174}\end{matrix}\right)$$

Extract the matrix elements a_{0} and a_{2}.

$$a_{0}=\frac{70816355}{112882},a_{2}=-\frac{62707}{790174}$$

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.

$$5a_{0}+5327a_{2}=2714,157a_{0}+9233a_{2}=97761$$

To make 5a_{0} and 157a_{0} equal, multiply all terms on each side of the first equation by 157 and all terms on each side of the second by 5.

$$157\times 5a_{0}+157\times 5327a_{2}=157\times 2714,5\times 157a_{0}+5\times 9233a_{2}=5\times 97761$$

Simplify.

$$785a_{0}+836339a_{2}=426098,785a_{0}+46165a_{2}=488805$$

Subtract 785a_{0}+46165a_{2}=488805 from 785a_{0}+836339a_{2}=426098 by subtracting like terms on each side of the equal sign.

$$785a_{0}-785a_{0}+836339a_{2}-46165a_{2}=426098-488805$$

Add 785a_{0} to -785a_{0}. Terms 785a_{0} and -785a_{0} cancel out, leaving an equation with only one variable that can be solved.

$$836339a_{2}-46165a_{2}=426098-488805$$

Add 836339a_{2} to -46165a_{2}.

$$790174a_{2}=426098-488805$$

Add 426098 to -488805.

$$790174a_{2}=-62707$$

Divide both sides by 790174.

$$a_{2}=-\frac{62707}{790174}$$

Substitute -\frac{62707}{790174} for a_{2} in 157a_{0}+9233a_{2}=97761. Because the resulting equation contains only one variable, you can solve for a_{0} directly.

$$157a_{0}+9233\left(-\frac{62707}{790174}\right)=97761$$

Multiply 9233 times -\frac{62707}{790174}.

$$157a_{0}-\frac{82710533}{112882}=97761$$

Add \frac{82710533}{112882} to both sides of the equation.

$$157a_{0}=\frac{11118167735}{112882}$$

Divide both sides by 157.

$$a_{0}=\frac{70816355}{112882}$$

The system is now solved.

$$a_{0}=\frac{70816355}{112882},a_{2}=-\frac{62707}{790174}$$