Solve {c}{a=b-10}{7a+7b=462}

$$\left. \begin{array} { c } { a = b - 10 } \\ { 7 a + 7 b = 462 } \end{array} \right.$$

Solve for a, b

$$a=28$$

$$b=38$$

Steps Using Substitution

Consider the first equation. Subtract b from both sides.

$$a-b=-10$$

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.

$$a-b=-10,7a+7b=462$$

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

$$a-b=-10$$

Add b to both sides of the equation.

$$a=b-10$$

Substitute b-10 for a in the other equation, 7a+7b=462.

$$7\left(b-10\right)+7b=462$$

Multiply 7 times b-10.

$$7b-70+7b=462$$

Add 7b to 7b.

$$14b-70=462$$

Add 70 to both sides of the equation.

$$14b=532$$

Divide both sides by 14.

$$b=38$$

Substitute 38 for b in a=b-10. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=38-10$$

Add -10 to 38.

$$a=28$$

The system is now solved.

$$a=28,b=38$$

Steps Using Matrices

Consider the first equation. Subtract b from both sides.

$$a-b=-10$$

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

$$a-b=-10,7a+7b=462$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-1\\7&7\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-10\\462\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-1\\7&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\7&7\end{matrix}\right))\left(\begin{matrix}1&-1\\7&7\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\7&7\end{matrix}\right))\left(\begin{matrix}-10\\462\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\7&7\end{matrix}\right))\left(\begin{matrix}-10\\462\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\7&7\end{matrix}\right))\left(\begin{matrix}-10\\462\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-\left(-7\right)}&-\frac{-1}{7-\left(-7\right)}\\-\frac{7}{7-\left(-7\right)}&\frac{1}{7-\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}-10\\462\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{14}\\-\frac{1}{2}&\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}-10\\462\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-10\right)+\frac{1}{14}\times 462\\-\frac{1}{2}\left(-10\right)+\frac{1}{14}\times 462\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}28\\38\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=28,b=38$$

Steps Using Elimination

Consider the first equation. Subtract b from both sides.

$$a-b=-10$$

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.

$$a-b=-10,7a+7b=462$$

To make a 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 1.

$$7a+7\left(-1\right)b=7\left(-10\right),7a+7b=462$$

Simplify.

$$7a-7b=-70,7a+7b=462$$

Subtract 7a+7b=462 from 7a-7b=-70 by subtracting like terms on each side of the equal sign.

$$7a-7a-7b-7b=-70-462$$

Add 7a to -7a. Terms 7a and -7a cancel out, leaving an equation with only one variable that can be solved.

$$-7b-7b=-70-462$$

Add -7b to -7b.

$$-14b=-70-462$$

Add -70 to -462.

$$-14b=-532$$

Divide both sides by -14.

$$b=38$$

Substitute 38 for b in 7a+7b=462. Because the resulting equation contains only one variable, you can solve for a directly.

$$7a+7\times 38=462$$

Multiply 7 times 38.

$$7a+266=462$$

Subtract 266 from both sides of the equation.

$$7a=196$$

Divide both sides by 7.

$$a=28$$

The system is now solved.

$$a=28,b=38$$