Solve {l}{4a+b/2=-5}{4a-b/4=-7/2}

$$\left. \begin{array} { l } { 4 a + \frac { b } { 2 } = - 5 } \\ { 4 a - \frac { b } { 4 } = - \frac { 7 } { 2 } } \end{array} \right.$$

Solve for a, b

$$a=-1$$

$$b=-2$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 2.

$$8a+b=-10$$

Consider the second equation. Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$16a-b=-14$$

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.

$$8a+b=-10,16a-b=-14$$

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

$$8a+b=-10$$

Subtract b from both sides of the equation.

$$8a=-b-10$$

Divide both sides by 8.

$$a=\frac{1}{8}\left(-b-10\right)$$

Multiply \frac{1}{8} times -b-10.

$$a=-\frac{1}{8}b-\frac{5}{4}$$

Substitute -\frac{b}{8}-\frac{5}{4} for a in the other equation, 16a-b=-14.

$$16\left(-\frac{1}{8}b-\frac{5}{4}\right)-b=-14$$

Multiply 16 times -\frac{b}{8}-\frac{5}{4}.

$$-2b-20-b=-14$$

Add -2b to -b.

$$-3b-20=-14$$

Add 20 to both sides of the equation.

$$-3b=6$$

Divide both sides by -3.

$$b=-2$$

Substitute -2 for b in a=-\frac{1}{8}b-\frac{5}{4}. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=-\frac{1}{8}\left(-2\right)-\frac{5}{4}$$

Multiply -\frac{1}{8} times -2.

$$a=\frac{1-5}{4}$$

Add -\frac{5}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a=-1$$

The system is now solved.

$$a=-1,b=-2$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 2.

$$8a+b=-10$$

Consider the second equation. Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$16a-b=-14$$

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

$$8a+b=-10,16a-b=-14$$

Write the equations in matrix form.

$$\left(\begin{matrix}8&1\\16&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-10\\-14\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}8&1\\16&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&1\\16&-1\end{matrix}\right))\left(\begin{matrix}8&1\\16&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\16&-1\end{matrix}\right))\left(\begin{matrix}-10\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\16&-1\end{matrix}\right))\left(\begin{matrix}-10\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\16&-1\end{matrix}\right))\left(\begin{matrix}-10\\-14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{8\left(-1\right)-16}&-\frac{1}{8\left(-1\right)-16}\\-\frac{16}{8\left(-1\right)-16}&\frac{8}{8\left(-1\right)-16}\end{matrix}\right)\left(\begin{matrix}-10\\-14\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{24}&\frac{1}{24}\\\frac{2}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}-10\\-14\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{24}\left(-10\right)+\frac{1}{24}\left(-14\right)\\\frac{2}{3}\left(-10\right)-\frac{1}{3}\left(-14\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-1\\-2\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=-1,b=-2$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 2.

$$8a+b=-10$$

Consider the second equation. Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$16a-b=-14$$

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.

$$8a+b=-10,16a-b=-14$$

To make 8a 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 8.

$$16\times 8a+16b=16\left(-10\right),8\times 16a+8\left(-1\right)b=8\left(-14\right)$$

Simplify.

$$128a+16b=-160,128a-8b=-112$$

Subtract 128a-8b=-112 from 128a+16b=-160 by subtracting like terms on each side of the equal sign.

$$128a-128a+16b+8b=-160+112$$

Add 128a to -128a. Terms 128a and -128a cancel out, leaving an equation with only one variable that can be solved.

$$16b+8b=-160+112$$

Add 16b to 8b.

$$24b=-160+112$$

Add -160 to 112.

$$24b=-48$$

Divide both sides by 24.

$$b=-2$$

Substitute -2 for b in 16a-b=-14. Because the resulting equation contains only one variable, you can solve for a directly.

$$16a-\left(-2\right)=-14$$

Subtract 2 from both sides of the equation.

$$16a=-16$$

Divide both sides by 16.

$$a=-1$$

The system is now solved.

$$a=-1,b=-2$$