Solve {l}{a+b=5}{2*572*a+890*b=3695}

$$\left\{ \begin{array} { l } { a + b = 5 } \\ { 2 \times 572 \times a + 890 \times b = 3695 } \end{array} \right.$$

Solve for a, b

$$a = -\frac{755}{254} = -2\frac{247}{254} \approx -2.972440945$$

$$b = \frac{2025}{254} = 7\frac{247}{254} \approx 7.972440945$$

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.

$$a+b=5,1144a+890b=3695$$

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

$$a+b=5$$

Subtract b from both sides of the equation.

$$a=-b+5$$

Substitute -b+5 for a in the other equation, 1144a+890b=3695.

$$1144\left(-b+5\right)+890b=3695$$

Multiply 1144 times -b+5.

$$-1144b+5720+890b=3695$$

Add -1144b to 890b.

$$-254b+5720=3695$$

Subtract 5720 from both sides of the equation.

$$-254b=-2025$$

Divide both sides by -254.

$$b=\frac{2025}{254}$$

Substitute \frac{2025}{254} for b in a=-b+5. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=-\frac{2025}{254}+5$$

Add 5 to -\frac{2025}{254}.

$$a=-\frac{755}{254}$$

The system is now solved.

$$a=-\frac{755}{254},b=\frac{2025}{254}$$

Steps Using Matrices

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

$$a+b=5,1144a+890b=3695$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\1144&890\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}5\\3695\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\1144&890\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\1144&890\end{matrix}\right))\left(\begin{matrix}1&1\\1144&890\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1144&890\end{matrix}\right))\left(\begin{matrix}5\\3695\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1144&890\end{matrix}\right))\left(\begin{matrix}5\\3695\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1144&890\end{matrix}\right))\left(\begin{matrix}5\\3695\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{890}{890-1144}&-\frac{1}{890-1144}\\-\frac{1144}{890-1144}&\frac{1}{890-1144}\end{matrix}\right)\left(\begin{matrix}5\\3695\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{445}{127}&\frac{1}{254}\\\frac{572}{127}&-\frac{1}{254}\end{matrix}\right)\left(\begin{matrix}5\\3695\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{445}{127}\times 5+\frac{1}{254}\times 3695\\\frac{572}{127}\times 5-\frac{1}{254}\times 3695\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{755}{254}\\\frac{2025}{254}\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=-\frac{755}{254},b=\frac{2025}{254}$$

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.

$$a+b=5,1144a+890b=3695$$

To make a and 1144a equal, multiply all terms on each side of the first equation by 1144 and all terms on each side of the second by 1.

$$1144a+1144b=1144\times 5,1144a+890b=3695$$

Simplify.

$$1144a+1144b=5720,1144a+890b=3695$$

Subtract 1144a+890b=3695 from 1144a+1144b=5720 by subtracting like terms on each side of the equal sign.

$$1144a-1144a+1144b-890b=5720-3695$$

Add 1144a to -1144a. Terms 1144a and -1144a cancel out, leaving an equation with only one variable that can be solved.

$$1144b-890b=5720-3695$$

Add 1144b to -890b.

$$254b=5720-3695$$

Add 5720 to -3695.

$$254b=2025$$

Divide both sides by 254.

$$b=\frac{2025}{254}$$

Substitute \frac{2025}{254} for b in 1144a+890b=3695. Because the resulting equation contains only one variable, you can solve for a directly.

$$1144a+890\times \frac{2025}{254}=3695$$

Multiply 890 times \frac{2025}{254}.

$$1144a+\frac{901125}{127}=3695$$

Subtract \frac{901125}{127} from both sides of the equation.

$$1144a=-\frac{431860}{127}$$

Divide both sides by 1144.

$$a=-\frac{755}{254}$$

The system is now solved.

$$a=-\frac{755}{254},b=\frac{2025}{254}$$