# How do I solve for Z ? (Interpolation)

#### Nikita.A

##### New member
Hello!
I am using interpolation to find a specific Z-Score for statistics. My math skills are a bit rusty, which is why I am having trouble solving for Z. Any help is appreciated. Thanks!

#### Mr. Bland

##### Junior Member
When a variable is buried within an expression like this, you "dig it out" piece by piece by moving everything else to the other side of the equals sign. Do keep in mind that both sides must remain equal at all times, so if you perform an operation to one side, you must perform the same operation to the other side.

We begin with this:

$$\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{2.2 - 2.1}$$​

Observe that if $$\displaystyle a - b = c$$, then $$\displaystyle a - c = b$$. We can adjust the equation as follows:

$$\displaystyle \frac{z - 2.1}{2.2 - 2.1} = 0.03573 - 0.035$$​

This has moved one of the terms away from $$\displaystyle z$$, to the other side of the equals sign.

Next, we will need to get $$\displaystyle z$$ out of the fraction. What happens if we multiply both sides of the equation by $$\displaystyle (2.2 - 2.1)$$?

#### Jomo

##### Elite Member
Observe that if $$\displaystyle a - b = c$$, then $$\displaystyle a - c = b$$. We can adjust the equation as follows:
$$\displaystyle \frac{z - 2.1}{2.2 - 2.1} = 0.03573 - 0.035$$
Yep!

#### Otis

##### Senior Member
I'm thinking the factor (0.03573 - 0.02781) is part of the equation.

I would have simplified the two subtractions that can be evaluated, first.

$$\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{0.1} \; (0.007920)$$

$$\;$$

#### Nikita.A

##### New member
When a variable is buried within an expression like this, you "dig it out" piece by piece by moving everything else to the other side of the equals sign. Do keep in mind that both sides must remain equal at all times, so if you perform an operation to one side, you must perform the same operation to the other side.

We begin with this:

$$\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{2.2 - 2.1}$$​

Observe that if $$\displaystyle a - b = c$$, then $$\displaystyle a - c = b$$. We can adjust the equation as follows:

$$\displaystyle \frac{z - 2.1}{2.2 - 2.1} = 0.03573 - 0.035$$​

This has moved one of the terms away from $$\displaystyle z$$, to the other side of the equals sign.

Next, we will need to get $$\displaystyle z$$ out of the fraction. What happens if we multiply both sides of the equation by $$\displaystyle (2.2 - 2.1)$$?
Thank you for your reply. If we multiply on both sides, it cancels out on the left and we add it to the right part of the equation.
Note: I think my picture (equation) was a bit unclear. The numbers in the bracket to the right are indeed part of the whole equation. Sorry for that!

#### Nikita.A

##### New member
I'm thinking the factor (0.03573 - 0.02781) is part of the equation.

I would have simplified the subtractions that can be evaluated, first.

$$\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{0.1} \; (0.007920)$$

$$\;$$
Yes that is part of the equation. It is also the part that I found to be the most complicated. Do I divide by 0.007920 to get it on the other side ? Thanks for your help!

#### Mr. Bland

##### Junior Member
Note: I think my picture (equation) was a bit unclear. The numbers in the bracket to the right are indeed part of the whole equation. Sorry for that!
No problem, it just becomes this instead:

$$\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{2.2 - 2.1}(0.03573 - 0.02781)$$​

And the same first step is performed:

$$\displaystyle \frac{z - 2.1}{2.2 - 2.1}(0.03573 - 0.02781) = 0.03573 - 0.035$$​

Thank you for your reply. If we multiply on both sides, it cancels out on the left and we add it to the right part of the equation.
Specifically, we multiply it on the right. I think by "add" you meant something like "put", but it's an important distinction!

Since there's an extra multiplication this time (after the correction), the next step is to divide both sides by $$\displaystyle (0.03573 - 0.02781)$$. Here's what that looks like:

$$\displaystyle \frac{z - 2.1}{2.2 - 2.1} = \frac{0.03573 - 0.035}{0.03573 - 0.02781}$$​

Then we multiply by $$\displaystyle (2.2 - 2.1)$$:

$$\displaystyle z - 2.1 = \frac{0.03573 - 0.035}{0.03573 - 0.02781}(2.2 - 2.1)$$​

Nearly there now! Only one step remains.

Since the numbers are a bit scattered, this is the basic form of the alegbra:

$$\displaystyle a = b - c\frac{z - d}{e}$$​

$$\displaystyle c\frac{z - d}{e} = b - a$$​

$$\displaystyle \frac{z - d}{e} = \frac{b - a}{c}$$​

$$\displaystyle z - d = e\frac{b - a}{c}$$​

$$\displaystyle z = e\frac{b - a}{c} + d$$​

#### Nikita.A

##### New member
Great
No problem, it just becomes this instead:

$$\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{2.2 - 2.1}(0.03573 - 0.02781)$$​

And the same first step is performed:

$$\displaystyle \frac{z - 2.1}{2.2 - 2.1}(0.03573 - 0.02781) = 0.03573 - 0.035$$​

Specifically, we multiply it on the right. I think by "add" you meant something like "put", but it's an important distinction!

Since there's an extra multiplication this time (after the correction), the next step is to divide both sides by $$\displaystyle (0.03573 - 0.02781)$$. Here's what that looks like:

$$\displaystyle \frac{z - 2.1}{2.2 - 2.1} = \frac{0.03573 - 0.035}{0.03573 - 0.02781}$$​

Then we multiply by $$\displaystyle (2.2 - 2.1)$$:

$$\displaystyle z - 2.1 = \frac{0.03573 - 0.035}{0.03573 - 0.02781}(2.2 - 2.1)$$​

Nearly there now! Only one step remains.

Since the numbers are a bit scattered, this is the basic form of the alegbra:

$$\displaystyle a = b - c\frac{z - d}{e}$$​

$$\displaystyle c\frac{z - d}{e} = b - a$$​

$$\displaystyle \frac{z - d}{e} = \frac{b - a}{c}$$​

$$\displaystyle z - d = e\frac{b - a}{c}$$​

$$\displaystyle z = e\frac{b - a}{c} + d$$​
Great, thank you for the detailed walkthrough. That sure helps me a lot. Now, I assume we just add +2.1 on both sides, to have z alone. After that, can I just put it into the calculator?

#### Cubist

##### Junior Member
If you post your answer then I'm sure someone here will check it.

(You can also check it yourself by putting your z value back into the RHS (Right Hand Side) of your original equation, using your calculator, and you should obtain the LHS value.

#### Nikita.A

##### New member
Great

Great, thank you for the detailed walkthrough. That sure helps me a lot. Now, I assume we just add +2.1 on both sides, to have z alone. After that, can I just put it into the calculator?
The Z-Score is 2.109. This turned out to be correct .

#### Otis

##### Senior Member
… [the factor (0.03573 - 0.02781) appearing on the right-hand side] is also the part that I found to be the most complicated.
Hi Nikita. You mean the most complicated to deal with, yes? Otherwise, it's just an expression for the number 0.00792.

(Note: I'm not sure why I wrote a zero at the end of 0.00792, in my previous reply. That was unnecessary.)

When I look at that number, expressed as (0.03573 - 0.02781), I also see it as a factor.

That is, the product is (z-2.1)/0.1 × 0.00792, and the factors are (z-2.1)/0.1 and 0.00792.

Do I divide by 0.007920 to get it on the other side? …
The phrase "get it on the other side" is not technically correct because it usually refers to adding (or subtracting) a term. Perhaps, you were thinking that division is the way to "eliminate" the factor 0.00792, on the right-hand side. If that's what you were thinking, then, yes, division could be one way to proceed. Just remember that when you divide a sum or difference by something, you must divide each term. We have

$$\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{0.1} \cdot 0.00792$$

There is one term on the left-hand side, and there are two terms on the right-hand side: 0.03573 and the product (z-2.1)/0.1×0.00792. We divide each term:

$$\displaystyle \frac{0.035}{0.00792} = \frac{0.03573}{0.00792} - \frac{z - 2.1}{0.1} \cdot \frac{0.00792}{0.00792}$$

$$\displaystyle 4.41919 = 4.51136 - \frac{z - 2.1}{0.1}$$

To solve for z, (1) subtract 4.51136 from each side and simplify, (2) multiply each side by -0.1 and simplify and (3) add 2.1 to each side and simplify.

This thread shows that we can solve for z using different steps. Each way is valid.

For those interested in seeing yet another set of steps, here's my approach. After evaluating (2.2-2.1) and (0.03573 - 0.02781), we have

$$\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{0.1} \cdot 0.00792$$

Subtract 0.03573 from each side and simplify

$$\displaystyle -0.00073 = -\frac{z - 2.1}{0.1} \cdot 0.00792$$

The product on the right-hand side may be viewed as

$$\displaystyle -0.00073 = (z - 2.1) \cdot \frac{-0.00792}{0.1}$$

Multiply each side by (-0.1)/(0.00792) and simplify

$$\displaystyle 0.009217 = z - 2.1$$

Add 2.1 to each side and simplify

$$\displaystyle 2.109217 = z$$

A couple notes about equivalent forms (shown symbolically, below). We may rearrange the product of a ratio times a number:

$$\displaystyle \frac{A}{B} \cdot C \quad=\quad \frac{A \cdot C}{B} \quad=\quad A \cdot \frac{C}{B}$$

When a negation symbol appears in front of a ratio, we may move it to the numerator or denominator

$$\displaystyle -\frac{a}{b} \quad=\quad \frac{-a}{b} \quad=\quad \frac{a}{-b}$$

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