Solve {l}{1/4a+1/2b+6=5/2}{16a+4b+6=6}

$$\left\{ \begin{array} { l } { \frac { 1 } { 4 } a + \frac { 1 } { 2 } b + 6 = \frac { 5 } { 2 } } \\ { 16 a + 4 b + 6 = 6 } \end{array} \right.$$

Solve for a, b

$$a=2$$

$$b=-8$$

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.

$$\frac{1}{4}a+\frac{1}{2}b+6=\frac{5}{2},16a+4b+6=6$$

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

$$\frac{1}{4}a+\frac{1}{2}b+6=\frac{5}{2}$$

Subtract 6 from both sides of the equation.

$$\frac{1}{4}a+\frac{1}{2}b=-\frac{7}{2}$$

Subtract \frac{b}{2} from both sides of the equation.

$$\frac{1}{4}a=-\frac{1}{2}b-\frac{7}{2}$$

Multiply both sides by 4.

$$a=4\left(-\frac{1}{2}b-\frac{7}{2}\right)$$

Multiply 4 times \frac{-b-7}{2}.

$$a=-2b-14$$

Substitute -2b-14 for a in the other equation, 16a+4b+6=6.

$$16\left(-2b-14\right)+4b+6=6$$

Multiply 16 times -2b-14.

$$-32b-224+4b+6=6$$

Add -32b to 4b.

$$-28b-224+6=6$$

Add -224 to 6.

$$-28b-218=6$$

Add 218 to both sides of the equation.

$$-28b=224$$

Divide both sides by -28.

$$b=-8$$

Substitute -8 for b in a=-2b-14. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=-2\left(-8\right)-14$$

Multiply -2 times -8.

$$a=16-14$$

Add -14 to 16.

$$a=2$$

The system is now solved.

$$a=2,b=-8$$

Steps Using Matrices

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

$$\frac{1}{4}a+\frac{1}{2}b+6=\frac{5}{2},16a+4b+6=6$$

Write the equations in matrix form.

$$\left(\begin{matrix}\frac{1}{4}&\frac{1}{2}\\16&4\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{2}\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}\frac{1}{4}&\frac{1}{2}\\16&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{2}\\16&4\end{matrix}\right))\left(\begin{matrix}\frac{1}{4}&\frac{1}{2}\\16&4\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{2}\\16&4\end{matrix}\right))\left(\begin{matrix}-\frac{7}{2}\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{2}\\16&4\end{matrix}\right))\left(\begin{matrix}-\frac{7}{2}\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{4}&\frac{1}{2}\\16&4\end{matrix}\right))\left(\begin{matrix}-\frac{7}{2}\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{4}{\frac{1}{4}\times 4-\frac{1}{2}\times 16}&-\frac{\frac{1}{2}}{\frac{1}{4}\times 4-\frac{1}{2}\times 16}\\-\frac{16}{\frac{1}{4}\times 4-\frac{1}{2}\times 16}&\frac{\frac{1}{4}}{\frac{1}{4}\times 4-\frac{1}{2}\times 16}\end{matrix}\right)\left(\begin{matrix}-\frac{7}{2}\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}&\frac{1}{14}\\\frac{16}{7}&-\frac{1}{28}\end{matrix}\right)\left(\begin{matrix}-\frac{7}{2}\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}\left(-\frac{7}{2}\right)\\\frac{16}{7}\left(-\frac{7}{2}\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}2\\-8\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=2,b=-8$$

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.

$$\frac{1}{4}a+\frac{1}{2}b+6=\frac{5}{2},16a+4b+6=6$$

To make \frac{a}{4} and 16a equal, multiply all terms on each side of the first equation by 16 and all terms on each side of the second by \frac{1}{4}.

$$16\times \frac{1}{4}a+16\times \frac{1}{2}b+16\times 6=16\times \frac{5}{2},\frac{1}{4}\times 16a+\frac{1}{4}\times 4b+\frac{1}{4}\times 6=\frac{1}{4}\times 6$$

Simplify.

$$4a+8b+96=40,4a+b+\frac{3}{2}=\frac{3}{2}$$

Subtract 4a+b+\frac{3}{2}=\frac{3}{2} from 4a+8b+96=40 by subtracting like terms on each side of the equal sign.

$$4a-4a+8b-b+96-\frac{3}{2}=40-\frac{3}{2}$$

Add 4a to -4a. Terms 4a and -4a cancel out, leaving an equation with only one variable that can be solved.

$$8b-b+96-\frac{3}{2}=40-\frac{3}{2}$$

Add 8b to -b.

$$7b+96-\frac{3}{2}=40-\frac{3}{2}$$

Add 96 to -\frac{3}{2}.

$$7b+\frac{189}{2}=40-\frac{3}{2}$$

Add 40 to -\frac{3}{2}.

$$7b+\frac{189}{2}=\frac{77}{2}$$

Subtract \frac{189}{2} from both sides of the equation.

$$7b=-56$$

Divide both sides by 7.

$$b=-8$$

Substitute -8 for b in 16a+4b+6=6. Because the resulting equation contains only one variable, you can solve for a directly.

$$16a+4\left(-8\right)+6=6$$

Multiply 4 times -8.

$$16a-32+6=6$$

Add -32 to 6.

$$16a-26=6$$

Add 26 to both sides of the equation.

$$16a=32$$

Divide both sides by 16.

$$a=2$$

The system is now solved.

$$a=2,b=-8$$