The article “Evaluating Vent Manifold Inerting Requirements: Flash Point Modelin

Step 1
Let ${\displaystyle x_1}$ represent acetic acid, ${\displaystyle x_2}$ represent proportionic acid and ${\displaystyle x_3}$ represent butyric acid.
We note that the predictors are Constant, Acetic Acid, Propionic Acid, Butyric Acid and the product of butyric acid and acetic acid.
${\displaystyle y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_1{x}_3}$
The estimates of ${\displaystyle {\beta }_0,\cdots ,{\beta }_4}$ are given in the column "SE Coef" of the Minitab output, which then results in the regression equation:
${\displaystyle \hat{y}=267.53-1.5926x_1-1.3897x_2-1.0934x_3-0.002658x_1{x}_3}$
Evaluate the regression equation at
${\displaystyle x_1=30}$
${\displaystyle x_2=35}$
${\displaystyle x_3=35}$
${\displaystyle \hat{y}=267.53-1.5926\left(30\right)-1.3897\left(35\right)-1.0934\left(30\right)-0.002658\left(30\right)\left(30\right)}$
Thus the predicted flash point is ${135.9183}^{\circ }F$
Step 2
Let ${\displaystyle x_1}$ represent acetic acid, ${\displaystyle x_2}$ represent proportionic acid and ${\displaystyle x_3}$ represent butyric acid.
We note that the predictors are Constant, Acetic Acid, Propionic Acid, Butyric Acid and the product of butyric acid and acetic acid.
${\displaystyle y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_1{x}_3}$
The acetic acid is represented bu ${\displaystyle x_1}$. Since ${\displaystyle x_1}$ occurs in the interaction term ${\displaystyle {\beta }_4{x}_1{x}_3}$ while the value of ${\displaystyle x_3}$ is unknown, we cannot determine the change in the predicted flash point ${\displaystyle \hat{y}}$ when the value of ${\displaystyle x_1}$ changes.
Step 3
Let ${\displaystyle x_1}$ represent acetic acid, ${\displaystyle x_2}$ represent proportionic acid and ${\displaystyle x_3}$ represent butyric acid.
We note that the predictors are Constant, Acetic Acid, Propionic Acid, Butyric Acid and the product of butyric acid and acetic acid.
${\displaystyle y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_1{x}_3}$
We are intersted in the propionic acid, which is represented by the variable ${\displaystyle x_2}$. Since ${\displaystyle x_2}$ does not occur in any interaction termsm it is possible to determine the change in the predicted y-values.
The slope with respect to Propionic acid is given in the row "Propionic acid" and in the column "Coef" of the given MINITAB output:
${\displaystyle \widehat{{\beta }_2}=-1.3897}$
When the x-values corresponding to Propionic acid differ by 10, then the predicted y-values will differ by 10 times the slope with respect to Propionic acid.
${\displaystyle 10\widehat{{\beta }_2}=10(-1.3897)=-13.897}$
Thus the predicted flashpoints will differ by ${\displaystyle {13.897}^{\circ }F}$