Solved on Feb 03, 2024

Find real numbers $a$ and $b$ such that $(a+1)+(b-2)i=4+6i$ , where $a=\square$ and $b=\square$ .

STEP 1

Assumptions
1. The equation given is $(a+1)+(b-2)i = 4+6i$ .
2. The variable $a$ represents the real part of the complex number.
3. The variable $b$ represents the imaginary part of the complex number.
4. The real parts of the complex numbers must be equal.
5. The imaginary parts of the complex numbers must be equal.

STEP 2

Separate the real and imaginary parts of the complex numbers on both sides of the equation.
The left side of the equation has the real part $a+1$ and the imaginary part $b-2$ . The right side of the equation has the real part $4$ and the imaginary part $6$ .

STEP 3

Set the real parts of the complex numbers equal to each other to find the value of $a$ .
$a + 1 = 4$

STEP 4

Solve for $a$ by subtracting 1 from both sides of the equation.
$a = 4 - 1$

STEP 5

Calculate the value of $a$ .
$a = 3$

STEP 6

Set the imaginary parts of the complex numbers equal to each other to find the value of $b$ .
Since the imaginary unit $i$ is the same on both sides, we can ignore it for this step and just compare the coefficients.
$(b - 2) = 6$

STEP 7

Solve for $b$ by adding 2 to both sides of the equation.
$b = 6 + 2$

STEP 8

Calculate the value of $b$ .
$b = 8$
The real numbers $a$ and $b$ that satisfy the equation are $a = 3$ and $b = 8$ .
$a = 3$ $b = 8$

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Find real numbers $a$ and $b$ such that $(a+1)+(b-2)i=4+6i$ , where $a=\square$ and $b=\square$ .

STEP 1

STEP 2

Separate the real and imaginary parts of the complex numbers on both sides of the equation.
The left side of the equation has the real part $a+1$ and the imaginary part $b-2$ . The right side of the equation has the real part $4$ and the imaginary part $6$ .

STEP 3

Set the real parts of the complex numbers equal to each other to find the value of $a$ .
$a + 1 = 4$

STEP 4

Solve for $a$ by subtracting 1 from both sides of the equation.
$a = 4 - 1$

STEP 5

Calculate the value of $a$ .
$a = 3$

STEP 6

Set the imaginary parts of the complex numbers equal to each other to find the value of $b$ .
Since the imaginary unit $i$ is the same on both sides, we can ignore it for this step and just compare the coefficients.
$(b - 2) = 6$

STEP 7

Solve for $b$ by adding 2 to both sides of the equation.
$b = 6 + 2$

STEP 8

Calculate the value of $b$ .
$b = 8$
The real numbers $a$ and $b$ that satisfy the equation are $a = 3$ and $b = 8$ .
$a = 3$ $b = 8$

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Find real numbers aaa and bbb such that (a+1)+(b−2)i=4+6i(a+1)+(b-2)i=4+6i(a+1)+(b−2)i=4+6i, where a=□a=\squarea=□ and b=□b=\squareb=□.

STEP 1

STEP 2

Separate the real and imaginary parts of the complex numbers on both sides of the equation.The left side of the equation has the real part a+1a+1a+1 and the imaginary part b−2b-2b−2. The right side of the equation has the real part 444 and the imaginary part 666.

STEP 3

Set the real parts of the complex numbers equal to each other to find the value of aaa.a+1=4a + 1 = 4a+1=4

STEP 4

Solve for aaa by subtracting 1 from both sides of the equation.a=4−1a = 4 - 1a=4−1

STEP 5

Calculate the value of aaa.a=3a = 3a=3

STEP 6

Set the imaginary parts of the complex numbers equal to each other to find the value of bbb.Since the imaginary unit iii is the same on both sides, we can ignore it for this step and just compare the coefficients.(b−2)=6(b - 2) = 6(b−2)=6

STEP 7

Solve for bbb by adding 2 to both sides of the equation.b=6+2b = 6 + 2b=6+2

STEP 8

Calculate the value of bbb.b=8b = 8b=8The real numbers aaa and bbb that satisfy the equation are a=3a = 3a=3 and b=8b = 8b=8.a=3a = 3a=3 b=8b = 8b=8

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find real numbers $a$ and $b$ such that $(a+1)+(b-2)i=4+6i$ , where $a=\square$ and $b=\square$ .

Separate the real and imaginary parts of the complex numbers on both sides of the equation.
The left side of the equation has the real part $a+1$ and the imaginary part $b-2$ . The right side of the equation has the real part $4$ and the imaginary part $6$ .

Set the real parts of the complex numbers equal to each other to find the value of $a$ .
$a + 1 = 4$

Solve for $a$ by subtracting 1 from both sides of the equation.
$a = 4 - 1$

Calculate the value of $a$ .
$a = 3$

Set the imaginary parts of the complex numbers equal to each other to find the value of $b$ .
Since the imaginary unit $i$ is the same on both sides, we can ignore it for this step and just compare the coefficients.
$(b - 2) = 6$

Solve for $b$ by adding 2 to both sides of the equation.
$b = 6 + 2$

Calculate the value of $b$ .
$b = 8$
The real numbers $a$ and $b$ that satisfy the equation are $a = 3$ and $b = 8$ .
$a = 3$ $b = 8$