Solve {l}{4a+2b=0}{a+b=1-sqrt{2}}

$$\left\{ \begin{array} { l } { 4 a + 2 b = 0 } \\ { a + b = 1 - \sqrt { 2 } } \end{array} \right.$$

Solve for a, b

$$a=\sqrt{2}-1\approx 0.414213562$$

$$b=2-2\sqrt{2}\approx -0.828427125$$

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.

$$4a+2b=0,a+b=1-\sqrt{2}$$

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

$$4a+2b=0$$

Subtract 2b from both sides of the equation.

$$4a=-2b$$

Divide both sides by 4.

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

Multiply \frac{1}{4} times -2b.

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

Substitute -\frac{b}{2} for a in the other equation, a+b=1-\sqrt{2}.

$$-\frac{1}{2}b+b=1-\sqrt{2}$$

Add -\frac{b}{2} to b.

$$\frac{1}{2}b=1-\sqrt{2}$$

Multiply both sides by 2.

$$b=2-2\sqrt{2}$$

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

$$a=-\frac{1}{2}\left(2-2\sqrt{2}\right)$$

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

$$a=\sqrt{2}-1$$

The system is now solved.

$$a=\sqrt{2}-1,b=2-2\sqrt{2}$$

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.

$$4a+2b=0,a+b=1-\sqrt{2}$$

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

$$4a+2b=0,4a+4b=4\left(1-\sqrt{2}\right)$$

Simplify.

$$4a+2b=0,4a+4b=4-4\sqrt{2}$$

Subtract 4a+4b=4-4\sqrt{2} from 4a+2b=0 by subtracting like terms on each side of the equal sign.

$$4a-4a+2b-4b=4\sqrt{2}-4$$

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

$$2b-4b=4\sqrt{2}-4$$

Add 2b to -4b.

$$-2b=4\sqrt{2}-4$$

Divide both sides by -2.

$$b=2-2\sqrt{2}$$

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

$$a+2-2\sqrt{2}=1-\sqrt{2}$$

Subtract 2-2\sqrt{2} from both sides of the equation.

$$a=\sqrt{2}-1$$

The system is now solved.

$$a=\sqrt{2}-1,b=2-2\sqrt{2}$$