Equality of Complex Numbers
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Question 1 of 7
1. Question
Solve for `x` and `y`.
`5x-12i=9-4yi`
Correct
Great Work!
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Imaginary equations in the form `a+bi=0` include a real part `a` and an imaginary part `bi`.To solve for `x` and `y` in `5x-12i=9-4yi`, set the real parts equal to each other and set the imaginary parts equal to each other.Solve for `x`.`color(red)(5x)+(-12)i=` `color(red)(9)+(-4y)i` The real parts are `5x` and `9`. `color(red)(5x)=` `color(red)(9)` Set the real parts equal to each other. `(5x)/5=` `9/5` Divide both sides by `5`. `x=` `9/5` Solve for `y`.`5x color(blue)(-12i)` `=9color(blue)(-4yi)` The imaginary parts are `-12i` and `-4yi`. `color(blue)(-12)=` `color(blue)(-4y)` Set the imaginary parts equal to each other. `(-12)/(-4)=` `(-4y)/(-4)` Divide both sides by `-4`. `3=` `y` Reverse the equation. `y=` `3` `x=9/5` and `y=3` -
Question 2 of 7
2. Question
Solve for `x` and `y`.
`4x-1+(5-2y)i=3x+2+(y-1)i`
Correct
Great Work!
Incorrect
Imaginary equations in the form `a+bi=0` include a real part `a` and an imaginary part `bi`.To solve for `x` and `y` in `4x-1+(5-2y)i=3x+2+(y-1)i`, set the real parts equal to each other and set the imaginary parts equal to each other.Solve for `x`.`color(red)(4x-1)+(5-2y)i=` `color(red)(3x+2)+(y-1)i` The real parts are `4x-1` and `3x+2`. `color(red)(4x-1)=` `color(red)(3x+2)` Set the real parts equal to each other. `(4-3)x=` `(2+1)` Combine like terms. `x=` `3` Solve for `y`.`color(blue)((5-2y)i)` `=color(blue)((y-1)i)` The imaginary parts are `(5-2y)i` and `(y-1)i`. `color(blue)((5-2y))=` `color(blue)((y-1))` Set the imaginary parts equal to each other. `(-2-1)y=` `(-1-5)` Combine like terms. `(-3y)/(-3)=` `(-6)/(-3)` Divide both sides by `-3`. `y=` `2` `x=3` and `y=2` -
Question 3 of 7
3. Question
Solve for `x` and `y`.
`11+2x-4i=7x+5yi-4+6i`
Correct
Great Work!
Incorrect
Imaginary equations in the form `a+bi=0` include a real part `a` and an imaginary part `bi`.To solve for `x` and `y` in `11+2x-4i=7x+5yi-4+6i`, set the real parts equal to each other and set the imaginary parts equal to each other.Solve for `x`.`color(red)(11+2x)-4i=` `color(red)(7x-4)+(5y+6)i` The real parts are `11+2x` and `7x-4`. `color(red)(11+2x)=` `color(red)(7x-4)` Set the real parts equal to each other. `(2-7)x=` `(-4-11)` Combine like terms. `(-5x)/(-5)=` `(-15)/(-5)` Divide both sides by `-5`. `x=` `3` Solve for `y`.`11+2x color(blue)(-4i)` `=7x-4 color(blue)(+(5y+6)i)` The imaginary parts are `-4i` and `(5y+6)i`. `color(blue)((-4))=` `color(blue)((5y+6))` Set the imaginary parts equal to each other. `(-5)y=` `(6+4)` Combine like terms. `(-5y)/(-5)=` `(10)/(-5)` Divide both sides by `-3`. `y=` `-2` `x=3` and `y=-2` -
Question 4 of 7
4. Question
Solve for `x` and `y`.
`(x+4i)(1-i)=9+yi`
Correct
Great Work!
Incorrect
Imaginary equations in the form `a+bi=0` include a real part `a` and an imaginary part `bi`.To solve for `x` and `y` in `(x+4i)(1-i)=9+yi`, expand, simplify, and then set the real parts equal to each other and set the imaginary parts equal to each other.Expand and simplify the left-hand side of the equation.`color(blue)((x+4i)(1-i))=` `9+yi` Expand the left-hand side of the equation. `x-x\i+4i-4i^2=` `9+yi` Simplify. Remember that `i^2=-1`. `x-x\i+4i+4=` `9+yi` Group the real parts together `x` and `4`. `(x+4)+i(-x+4)=` `9+yi` Group the imaginary parts together `-x\i` and `4i`. Solve for `x`.`color(red)((x+4))+i(-x+4)=` `color(red)(9)+yi` The real parts are `x+4` and `9`. `color(red)(x+4)=` `color(red)(9)` Set the real parts equal to each other. `x+4-4=` `9-4` Subtract `4` from each side. `x=` `5` Solve for `y`.`(x+4)color(blue)(+i(-x+4))=` `9color(blue)(+yi)` The imaginary parts are `i(-x+4)` and `yi`. `color(blue)(i(-x+4))=` `color(blue)(yi)` Set the imaginary parts equal to each other. `(i(-x+4))/i=` `(yi)/i` Divide both sides by `i`. `-x+4=` `y` Sub in `x=5` `-(5)+4=` `y` Simplify `-1=` `y` Reverse the sides `y=` `-1` `x=5` and `y=-1` -
Question 5 of 7
5. Question
Solve for `x` and `y`.
`(2+2i)(x+yi)=4+12i`
Correct
Great Work!
Incorrect
Imaginary equations in the form `a+bi=0` include a real part `a` and an imaginary part `bi`.To solve for `x` and `y` in `(2+2i)(x+yi)=4+12i`, expand, simplify, and then set the real parts equal to each other and set the imaginary parts equal to each other.Expand and simplify the left-hand side of the equation.`color(blue)((2+2i)(x+yi))=` `4+12i` Expand the left-hand side of the equation. `2x+2yi+2xi+2yi^2=` `4+12i` Simplify. Remember that `i^2=-1`. `(2x-2y)+2yi+2xi=` `4+12i` Group the real parts together. `(2x-2y)+(2x+2y)i=` `4+12i` Group the imaginary parts together. Solve for `x`.`color(red)(2x-2y)+(2x+2y)i=` `color(red)(4)+12i` The real parts are `2x-2y` and `4`. `color(red)(2x-2y)=` `color(red)(4)` Set the real parts equal to each other. `2x-2y=` `4` Set as Equation 1. Solve for `y`.`2x-2y color(blue)(+(2x+2y)i)=` `4color(blue)(+12i)` The imaginary parts are `i(2x+2y)` and `12i`. `color(blue)(i(2x+2y))=` `color(blue)(12i)` Set the imaginary parts equal to each other. `(i(2x+2y))/i=` `(12i)/i` Divide both sides by `i`. `2x+2y=` `12` Set as Equation 2. Subtracting Equation 1 and Equation 2.`color(blue)(2x-2y)-color(red)(2x+2y)=` `color(blue)(4)-color(red)(12)` Subtract Equation 1 and Equation 2. `4y=` `8` Combine like terms `(4y)/4=` `8/4` Divide both sides by `4`. `y=` `2` Substitute `y=2` to Equation 2.`2x+2(2)=` `12` Substitute y = 2 to Equation 2. `2x=` `12-4` Combine like terms `(2x)/2=` `8/2` Divide both sides by `2`. `x=` `4` `x=4` and `y=2` -
Question 6 of 7
6. Question
Solve for `x` and `y`.
`(2x+yi)(1-i)=2i`
Correct
Great Work!
Incorrect
Imaginary equations in the form `a+bi=0` include a real part `a` and an imaginary part `bi`.To solve for `x` and `y` in `(2x+yi)(1-i)=2i`, expand, simplify, and then set the real parts equal to each other and set the imaginary parts equal to each other.Expand and simplify the left-hand side of the equation.`color(blue)((2x+yi)(1-i))=` `2i` Expand the left-hand side of the equation. `2x-2x\i+yi-yi^2=` `2i` Simplify. Remember that `i^2=-1`. `2x-2x\i+yi+y=` `2i` Group the real parts together. `(2x+y)+i(-2x+y)=` `2i` Group the imaginary parts together. Solve for `x`.`color(red)(2x+y)+(-2x+y)i=` `2i` The real parts are `2x+y` and `0`. `color(red)(2x+y)=` `0` Set the real parts equal to each other. `2x+y=` `0` Set as Equation 1. Solve for `y`.`2x+y color(blue)((-2x+y)i)=` `color(blue)(2i)` The imaginary parts are `i(-2x+y)` and `2i`. `color(blue)(i(-2x+y))=` `color(blue)(2i)` Set the imaginary parts equal to each other. `(i(-2x+y))/i=` `(2i)/i` Divide both sides by `i`. `-2x+y=` `2` Set as Equation 2. Adding Equation 1 and Equation 2.`color(blue)(2x+y)+color(red)(-2x+y)=` `color(blue)(0)+color(red)(2)` Subtract Equation 1 and Equation 2. `2y=` `2` Combine like terms `(2y)/2=` `2/2` Divide both sides by `4`. `y=` `1` Substitute `y=1` to Equation 1.`2x+1=` `0` Substitute y = 2 to Equation 2. `2x=` `-1` Combine like terms `(2x)/2=` `(-1)/2` Divide both sides by `2`. `x=` `(-1)/2` `x=(-1)/2` and `y=1` -
Question 7 of 7
7. Question
Solve for `x` and `y`.
`(x-yi)(3+i)=i`
Correct
Great Work!
Incorrect
Imaginary equations in the form `a+bi=0` include a real part `a` and an imaginary part `bi`.To solve for `x` and `y` in `(x-yi)(3+i)=i`, expand, simplify, and then set the real parts equal to each other and set the imaginary parts equal to each other.Expand and simplify the left-hand side of the equation.`color(blue)((x-yi)(3+i)=i)=` `i` Expand the left-hand side of the equation. `3x+x\i-3yi-yi^2=` `i` Simplify. Remember that `i^2=-1`. `(3x+y)+x\i-3yi=` `i` Group the real parts together. `(3x+y)+i(x-3y)=` `i` Group the imaginary parts together. Solve for `x`.`color(red)(3x+y)+(x-3y)i=` `i` The real parts are `3x+y` and `0`. `color(red)(3x+y)=` `0` Set the real parts equal to each other. `3x+y=` `0` Set as Equation 1. Solve for `y`.`3x + y color(blue)(+(x-3y)i)=` `color(blue)(i)` The imaginary parts are `(x-3y)i` and `i`. `color(blue)(i(x-3y))=` `color(blue)(i)` Set the imaginary parts equal to each other. `(i(x-3y))/i=` `i/i` Divide both sides by `i`. `x-3y=` `1` Set as Equation 2. Solving for `x`.`(3x+y)times3=` `0times3` Multiply the whole equation by 3. `9x+3y=` `0` Set as Equation 3. `color(blue)(x-3y)+color(red)(9x+3y)=` `color(blue)(1)+color(red)(0)` Add Equation 2 and Equation 3. `(1+9)x=` `1` Combine like terms. `(10x)/10=` `1/10` Divide both sides by `10`. `x=` `1/10` Solving for `y`.`3(1/10)+y=` `0` Substitute `x = 1/10` to Equation 1. `y=` `(-3)/10` Combine like terms. `y=` `(-3)/10` `x=1/10` and `y=(-3)/10`
Quizzes
- Simplify Roots of Negative Numbers 1
- Simplify Roots of Negative Numbers 2
- Powers of the Imaginary Unit 1
- Powers of the Imaginary Unit 2
- Solve Quadratic Equations with Complex Solutions 1
- Solve Quadratic Equations with Complex Solutions 2
- Equality of Complex Numbers
- Add and Subtract Complex Numbers 1
- Add and Subtract Complex Numbers 2
- Multiply Complex Numbers 1
- Multiply Complex Numbers 2
- Divide Complex Numbers
- Complex Numbers – Product of Linear Factors 1
- Complex Numbers – Product of Linear Factors 2
- Mixed Operations with Complex Numbers