To isolate a variable in an equation, you must remember your Order of Operations. But while with "plugging 'n chugging" in a regular equation you can use PEMDAS to remember order of operations to solve a problem, you have to work backwards for isolating a variable.
S.A.D.M.E.P: Subtraction, Addition, Division, Multiplication, Exponents, Parentheses,
Rules
The key to isolating variables is to get every multiple or quotient with that variable on one side of the equal side and everything without that variable on the other side. Finally, combine any terms that aren't already combined, and cancel the coefficient of the variable, if there is one. You may also need to factor out the desired variable.
As some math operations are transitive, there is usually more than one way to isolate a variable in any given equation. As long as you are following the laws of mathematics correctly, the answer should eventually be the same.
Simplify your answer as much as possible. There should NEVER be a fraction within a fraction. There should also never be a radical sign in the denominator only (only when an entire fraction is under a radical sign is a radical sign acceptable in this case). You should also never have negative exponents. Follow the laws of exponents for help on this topic.
This process can be VERY tricky, so practice A LOT!!!
Example 1:
Isolate for z in the following equation:
x = z/y
(y) x = z / y (y)
(y) x = z / y (y)
xy = z
Example 2:
Isolate for a in the following equation:
4c = 2a + 6b
4c - 6b = 2a + 6b - 6b
4c - 6b = 2a + 6b - 6b
4c - 6b = 2a
*Remember that the 2 must be distributed to both 4c & 6b, hence the parentheses.
(4c - 6b) / 2 = 2a / 2
(4c - 6b) / 2 = a
(4/2)c - (6/2)b = a
2c - 3b = a**
**You could have divided out the 2 from the very beginning and would have gotten the same answer. As long as you remember to divide 2 out from BOTH sides of the equation and to divide it from ALL number groupings.
Example 3:
Isolate for x in the following equation:
5x + 8 = 3x – 6
5x + 8 - 3x = 3x - 3x – 6
5x + 8 - 3x = 3x - 3x – 6
5x + 8 - 8 - 3x = 3x - 3x – 6 - 8
5x + 8 - 8 - 3x = 3x - 3x – 6 - 8
5x - 3x + 8 - 8 = 3x - 3x – 6 - 8
2x + 8 - 8 = 3x - 3x –14
2x = –14
2x / 2 = –14 / 2x = –7
Example 4:
Isolate for m in the following equation:
3s = (6n + 9m) / (2m - 5o)
3s (2m - 5o) = (6n + 9m) (2m - 5o)/ (2m - 5o)
3s (2m - 5o) = 6n + 9m
(3s)(2m) - (3s)(5o) = 6n + 9m
6ms - 15os = 6n + 9m
6ms - 15os - 9m = 6n + 9m - 9m
6ms - 15os + 15os - 9m= 6n + 15os + 9m - 9m
6ms - 15os + 15os - 9m = 6n + 15os + 9m - 9m
6ms - 9m = 6n + 15os
m(6s - 9) = 6n + 15os
m(6s - 9) = 6n + 15os
m (6s - 9) / (6s - 9) = (6n + 15os) / (6s - 9)
m = (6n + 15os) / (6s - 9)
3s (2m - 5o) / 3s = (6n + 9m) / 3s
3s (2m - 5o) / 3s = (6n + 9m) / 3s
2m - 5o = (6n + 9m) / 3s
2m - 5o = 2n/s + 3m/s
2m - 3m/s - 5o + 5o = 2n/s + 5o + 3m/s - 3m/s
2m - 3m/s - 5o + 5o = 2n/s+ 5o + 3m/s - 3m/s
2m - 3m/s = 2n/s + 5o
m(2 - 3/s) = 2n/s + 5o
m(2 - 3/s) = 2n/s + 5o
m (2 - 3/s) / (2 - 3/s) = (2n/s + 5o) / (2 - 3/s)m = (2n/s + 5o) / (2 - 3/s)
m = (2(s)n/s + 5(s)o) / (2(s) - 3(s)/s)
m = (2n + 5os) / (2s - 3)**
**Both paths will ultimately yield the same answer. Even if the answers do not initially look the same, that may be due to simplification or lack there of.**
S.A.D.M.E.P: Subtraction, Addition, Division, Multiplication, Exponents, Parentheses,
Rules
- Whatever is done to one side of the equation must be done on the other side as well to remain balanced.
- Do the opposite operation that is being done (ex: if the multiple or quotient is being divided in the equation, multiply it on both sides).
- Add or subtract all quotients not containing the variable to one side of the equation.*
- If the variable is in a denominator, simply the equation by multiplying by the reciprocal on both sides.
- Multiply or divide numbers, not the variable.*
- Exponents (or radicals) must be done next.
- If the variable was in parentheses and that has been isolated, the parentheses are deemed moot (remove them) and continue isolating starting at the beginning of SADMEP all over again.
The key to isolating variables is to get every multiple or quotient with that variable on one side of the equal side and everything without that variable on the other side. Finally, combine any terms that aren't already combined, and cancel the coefficient of the variable, if there is one. You may also need to factor out the desired variable.
As some math operations are transitive, there is usually more than one way to isolate a variable in any given equation. As long as you are following the laws of mathematics correctly, the answer should eventually be the same.
Simplify your answer as much as possible. There should NEVER be a fraction within a fraction. There should also never be a radical sign in the denominator only (only when an entire fraction is under a radical sign is a radical sign acceptable in this case). You should also never have negative exponents. Follow the laws of exponents for help on this topic.
This process can be VERY tricky, so practice A LOT!!!
Example 1:
Isolate for z in the following equation:
x = z/y
- Addition/Subtraction: There is no addition/subtraction in this problem, so next is dealing with multiplication/division.
- Multiplication/Division: To get z by itself, you need to move the y to the other side of the equal sign. Since it is being divided in the original equation, the opposite operation (multiplication) done on both sides of the equation is needed to move it to the left side of the equal sign.
(y) x = z / y (y)
(y) x = z / y (y)
xy = z
Example 2:
Isolate for a in the following equation:
4c = 2a + 6b
- Addition/Subtraction: We need to move 6b to the other side of the equation. Remember, everything with a on one side, everything without a on the other. Since 6b is added in the original equation, it must be subtracted on both sides.
4c - 6b = 2a + 6b - 6b
4c - 6b = 2a + 6b - 6b
4c - 6b = 2a
- Multiplication/Division: We need to move the 2 to the other side of the equation. Since it is being multiplied in the original, it must be divided on both sides.
*Remember that the 2 must be distributed to both 4c & 6b, hence the parentheses.
(4c - 6b) / 2 = 2a / 2
(4c - 6b) / 2 = a
- Simplify:
(4/2)c - (6/2)b = a
2c - 3b = a**
**You could have divided out the 2 from the very beginning and would have gotten the same answer. As long as you remember to divide 2 out from BOTH sides of the equation and to divide it from ALL number groupings.
Example 3:
Isolate for x in the following equation:
5x + 8 = 3x – 6
- Addition/Subtraction: Get every grouping with x on one side of the equation and every grouping without x on the other side. It makes no difference whether x ends up on the right or left of the equal sign. The final answer should still be the same.
5x + 8 - 3x = 3x - 3x – 6
5x + 8 - 3x = 3x - 3x – 6
5x + 8 - 8 - 3x = 3x - 3x – 6 - 8
5x + 8 - 8 - 3x = 3x - 3x – 6 - 8
5x - 3x + 8 - 8 = 3x - 3x – 6 - 8
2x + 8 - 8 = 3x - 3x –14
2x = –14
- Multiplication/Division: We need to move the 2 to the other side of the equation. Since it is being multiplied in the original, it must be divided on both sides.
2x / 2 = –14 / 2x = –7
Example 4:
Isolate for m in the following equation:
3s = (6n + 9m) / (2m - 5o)
- Addition/Subtraction: There is no addition/subtraction outside of the parentheses in this problem, so next is dealing with multiplication/division.
- Multiplication/Division: Any time you have the intended variable in the denominator, get it out of there as early in the problem as possible (following S.A.D.M.E.P. of course). Treat (2m-5o) as one grouping. Since it is being divided in the original, it must be multiplied on both sides.
3s (2m - 5o) = (6n + 9m) (2m - 5o)/ (2m - 5o)
3s (2m - 5o) = 6n + 9m
- At this point there are 2 different approaches that will ultimately yield the same result. One path is shorter than the other and a little easier, but you will only be able to be immediately aware of which it is based on lots of practice.
- Distribute: Distribute 3sto everything inside the parenthesis.
(3s)(2m) - (3s)(5o) = 6n + 9m
6ms - 15os = 6n + 9m
- Addition/Subtraction: With the disappearance of the parenthesis, we start at the beginning of S.A.D.M.E.P. again. We need to get every grouping with m on one side and every other grouping on the other side.
6ms - 15os - 9m = 6n + 9m - 9m
6ms - 15os + 15os - 9m= 6n + 15os + 9m - 9m
6ms - 15os + 15os - 9m = 6n + 15os + 9m - 9m
6ms - 9m = 6n + 15os
- Factor: This is a variation of multiplication/division, as we are dividing out mfrom both groupings on the left side of the equation. However we are NOT doing this to the right side. That is what makes it factoring, rather than dividing.
m(6s - 9) = 6n + 15os
m(6s - 9) = 6n + 15os
- Multiplication/Division: To isolate m, divide everything inside the parenthesis on bot sides of the equation. Many students want to redistribute the m they just factored out. This would erase the previous step.
m (6s - 9) / (6s - 9) = (6n + 15os) / (6s - 9)
m = (6n + 15os) / (6s - 9)
- Simplify:
- Divide: Divide3s on both sides of the equation in order to move it to the right side of the equal sign.
3s (2m - 5o) / 3s = (6n + 9m) / 3s
3s (2m - 5o) / 3s = (6n + 9m) / 3s
2m - 5o = (6n + 9m) / 3s
- Distribute: Distribute 3s to everything inside the parenthesis.
2m - 5o = 2n/s + 3m/s
- Addition/Subtraction: With the disappearance of the parenthesis, we start at the beginning of S.A.D.M.E.P. again. We need to get every grouping with mon one side and every other grouping on the other side.
2m - 3m/s - 5o + 5o = 2n/s + 5o + 3m/s - 3m/s
2m - 3m/s - 5o + 5o = 2n/s+ 5o + 3m/s - 3m/s
2m - 3m/s = 2n/s + 5o
- Factor: This is a variation of multiplication/division, as we are dividing out mfrom both groupings on the left side of the equation. However we are NOT doing this to the right side. That is what makes it factoring, rather than dividing.
m(2 - 3/s) = 2n/s + 5o
m(2 - 3/s) = 2n/s + 5o
- Multiplication/Division: To isolate m, divide everything inside the parenthesis on bot sides of the equation. Many students want to redistribute the m they just factored out. This would erase the previous step.
m (2 - 3/s) / (2 - 3/s) = (2n/s + 5o) / (2 - 3/s)m = (2n/s + 5o) / (2 - 3/s)
- Simplify: You can't have fractions within fractions. In this case, it is easy. As bot fractions involves, simply multiply s in the numerator and in the denominator of the dominant fraction.
m = (2(s)n/s + 5(s)o) / (2(s) - 3(s)/s)
m = (2n + 5os) / (2s - 3)**
**Both paths will ultimately yield the same answer. Even if the answers do not initially look the same, that may be due to simplification or lack there of.**
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