Solve (a+2b)+(-3a-5b)i=3-7i

$$( a + 2 b ) + ( - 3 a - 5 b ) i = 3 - 7 i$$

Solve for a

$$a=\left(-\frac{17}{10}-\frac{1}{10}i\right)b+\left(\frac{12}{5}+\frac{1}{5}i\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply -3a-5b by i.

$$a+2b-3ia-5ib=3-7i$$

Combine a and -3ia to get \left(1-3i\right)a.

$$\left(1-3i\right)a+2b-5ib=3-7i$$

Combine 2b and -5ib to get \left(2-5i\right)b.

$$\left(1-3i\right)a+\left(2-5i\right)b=3-7i$$

Subtract \left(2-5i\right)b from both sides.

$$\left(1-3i\right)a=3-7i-\left(2-5i\right)b$$

Multiply -1 and 2-5i to get -2+5i.

$$\left(1-3i\right)a=3-7i+\left(-2+5i\right)b$$

The equation is in standard form.

$$\left(1-3i\right)a=\left(-2+5i\right)b+\left(3-7i\right)$$

Divide both sides by 1-3i.

$$\frac{\left(1-3i\right)a}{1-3i}=\frac{\left(-2+5i\right)b+\left(3-7i\right)}{1-3i}$$

Dividing by 1-3i undoes the multiplication by 1-3i.

$$a=\frac{\left(-2+5i\right)b+\left(3-7i\right)}{1-3i}$$

Divide 3-7i+\left(-2+5i\right)b by 1-3i.

$$a=\left(-\frac{17}{10}-\frac{1}{10}i\right)b+\left(\frac{12}{5}+\frac{1}{5}i\right)$$

Solve for b

$$b=\left(-\frac{17}{29}+\frac{1}{29}i\right)a+\left(\frac{41}{29}+\frac{1}{29}i\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply -3a-5b by i.

$$a+2b-3ia-5ib=3-7i$$

Combine a and -3ia to get \left(1-3i\right)a.

$$\left(1-3i\right)a+2b-5ib=3-7i$$

Combine 2b and -5ib to get \left(2-5i\right)b.

$$\left(1-3i\right)a+\left(2-5i\right)b=3-7i$$

Subtract \left(1-3i\right)a from both sides.

$$\left(2-5i\right)b=3-7i-\left(1-3i\right)a$$

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

$$\left(2-5i\right)b=3-7i+\left(-1+3i\right)a$$

The equation is in standard form.

$$\left(2-5i\right)b=\left(-1+3i\right)a+\left(3-7i\right)$$

Divide both sides by 2-5i.

$$\frac{\left(2-5i\right)b}{2-5i}=\frac{\left(-1+3i\right)a+\left(3-7i\right)}{2-5i}$$

Dividing by 2-5i undoes the multiplication by 2-5i.

$$b=\frac{\left(-1+3i\right)a+\left(3-7i\right)}{2-5i}$$

Divide 3-7i+\left(-1+3i\right)a by 2-5i.

$$b=\left(-\frac{17}{29}+\frac{1}{29}i\right)a+\left(\frac{41}{29}+\frac{1}{29}i\right)$$