Solve (2a-b)+(3a+2b)i=-8+9i

$$( 2 a - b ) + ( 3 a + 2 b ) i = - 8 + 9 i$$

Solve for a

$$a=\left(-\frac{4}{13}-\frac{7}{13}i\right)b+\left(\frac{11}{13}+\frac{42}{13}i\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3a+2b by i.

$$2a-b+3ia+2ib=-8+9i$$

Combine 2a and 3ia to get \left(2+3i\right)a.

$$\left(2+3i\right)a-b+2ib=-8+9i$$

Combine -b and 2ib to get \left(-1+2i\right)b.

$$\left(2+3i\right)a+\left(-1+2i\right)b=-8+9i$$

Subtract \left(-1+2i\right)b from both sides.

$$\left(2+3i\right)a=-8+9i-\left(-1+2i\right)b$$

Multiply -1 and -1+2i to get 1-2i.

$$\left(2+3i\right)a=-8+9i+\left(1-2i\right)b$$

The equation is in standard form.

$$\left(2+3i\right)a=\left(1-2i\right)b+\left(-8+9i\right)$$

Divide both sides by 2+3i.

$$\frac{\left(2+3i\right)a}{2+3i}=\frac{\left(1-2i\right)b+\left(-8+9i\right)}{2+3i}$$

Dividing by 2+3i undoes the multiplication by 2+3i.

$$a=\frac{\left(1-2i\right)b+\left(-8+9i\right)}{2+3i}$$

Divide -8+9i+\left(1-2i\right)b by 2+3i.

$$a=\left(-\frac{4}{13}-\frac{7}{13}i\right)b+\left(\frac{11}{13}+\frac{42}{13}i\right)$$

Solve for b

$$b=\left(-\frac{4}{5}+\frac{7}{5}i\right)a+\left(\frac{26}{5}+\frac{7}{5}i\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3a+2b by i.

$$2a-b+3ia+2ib=-8+9i$$

Combine 2a and 3ia to get \left(2+3i\right)a.

$$\left(2+3i\right)a-b+2ib=-8+9i$$

Combine -b and 2ib to get \left(-1+2i\right)b.

$$\left(2+3i\right)a+\left(-1+2i\right)b=-8+9i$$

Subtract \left(2+3i\right)a from both sides.

$$\left(-1+2i\right)b=-8+9i-\left(2+3i\right)a$$

Multiply -1 and 2+3i to get -2-3i.

$$\left(-1+2i\right)b=-8+9i+\left(-2-3i\right)a$$

The equation is in standard form.

$$\left(-1+2i\right)b=\left(-2-3i\right)a+\left(-8+9i\right)$$

Divide both sides by -1+2i.

$$\frac{\left(-1+2i\right)b}{-1+2i}=\frac{\left(-2-3i\right)a+\left(-8+9i\right)}{-1+2i}$$

Dividing by -1+2i undoes the multiplication by -1+2i.

$$b=\frac{\left(-2-3i\right)a+\left(-8+9i\right)}{-1+2i}$$

Divide -8+9i+\left(-2-3i\right)a by -1+2i.

$$b=\left(-\frac{4}{5}+\frac{7}{5}i\right)a+\left(\frac{26}{5}+\frac{7}{5}i\right)$$