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Isolating variables

Isolating variables questions on the SAT

SAT Subscore: Passport to Advanced Mathematics

Isolated variable is a variable that is written alone on one side of the equation and the second side of the equation contains other variables. For example: In the equation y=mx+b the variable y is isolated.

Insulating versus solving: We can isolate any variable of the equation by rearranging the equation according to the same rules we use to solve it. These are the rules for maintaining equality: we can add or subtract the same value from each side of the equation and we can divide or multiply each side of the equation by a same value.

Note that the isolated variable will be equal to an expression containing other variables and not a numeric solution like after solving an equation.

Isolating variables questions may ask you to isolate variables in different types of equations, such as linear equations or quadratic equations.

If you are not familiar with solving an equation topic visit the page about linear equations before learning this topic.

Isolating variables in linear equations

You need to follow the rules for maintaining equality, in addition you may need to use distributing and combining like terms. If you need to learn more, see linear equations topic.

Consider the following example:

Slope intercept form of a linear function is given by the equation y=3x+7. What is x in terms of y?

y=3x+7
y-7=3x+7-7
y-7=3x
x=(y-7)/3

Now we can calculate x coordinate given y coordinate area. For example: if y coordinate is 16 then x coordinate is x=(16-7)/3=3.

Checking the answer: 
y=3x+7
y=3((y-7)/3)+7
y=y-7+7
y=y

Consider the following example:

5a2-7b=3a2-3b+5

What is b in terms of a?

5a2-7b=3a2-3b+5
-3b+7b=5a2-3a2-5
4b=2a2-5
b=(2a2-5)/4

Checking the answer:
5a2-7b=3a2-3b+5
5a2-7(2a2-5)/4=3a2-3(2a2-5)/4+5

Multiply each part of the equation by 4:
20a2-7(2a2-5) =12a2-3(2a2-5)+20
20a2-14a2+35=12a2-6a2+15+20
6a2+35=6a2+35

Consider the following example:

The plant uses function c defined by c(n)=100,000/n+5 to calculate the cost c(n), in dollars, of producing one product. The variable production cost is 5 dollars, the fixed cost of the plant is 100,000 dollars and n is defined as the quantity of the products.

Rewrite n in terms of c.

c=100,000/n+5
c-5=100,000/n+5-5
c-5=100,000/n
n(c-5)=100,000n/n
n(c-5)=100,000
n(c-5)/(c-5)=100,000/(c-5)
n=100,000/(c-5)

Now we can calculate the number of units that were produced given the production cost of a product. For example: if the production cost of a product is 10 dollars, what is the number of products that the plant produced?
n=100,000/(c-5)
n=100,000/(10-5)
n=100,000/5=20,000

Checking the answer:
c=100,000/n+5
c=100,000/(100,000/(c-5))+5
c=100,000*((c-5)/100,000)+5
c=c-5+5
c=c

Isolating variables in quadratic equations without x term

Since we have only x2 term and there is no x term in the equation, we can isolate x

Step 1: Isolate x2 with the rules for maintaining equality, in addition you may need to use distributing and combining like terms. If you need to learn more, see linear equations topic.

Step 2: Take a square root from each side of the equation. If you need to learn more, see quadratic equations topic.

Consider the following example:

The area of a square is the product of the length l of each side with itself A=l2.

What is l in terms of A?

Step 2: Take a square root of both sides of the equation:
A=l2
√ A=√ l2
√ A=l
l=√ A

Now we can calculate the length of each side of a square given its area. For example: if the area is 49 then the length of its side is √ 49=7.

Checking the answer:
A=l2
A=√ A 2
A=A

Consider the following example:

5a2-7b=3a2-3b+5

What is a equal to in terms of b?

Step 1: Isolate x
5a2-7b=a2-3b+5
5a2-a2=-3b+7b+5
4a2=4b+5
4a2/2=(4b+5)/2
a2=(4b+5)/4

Step 2: Take a square root from each side of the equation
a2=(4b+5)/4
√a2=√(4b+5)/√4
a=√(4b+5)/2

Checking the answer:
5a2-7b=a2-3b+5
5(√(4b+5)/2)2-7b=(√(4b+5)/2)2-3b+5
(5(4b+5))/4 -7b=(4b+5)/4-3b+5

Multiply each part of the equation by 4:
5(4b+5)-28b=4b+5-12b+20
20b+25-28b=4b+5-12b+20
-8b+25=-8b+25

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