# Hypothesis test for multiple regression

Consider the following model $$y_i = \beta_0+ \beta_1x_{i1}+ \beta_2x_{i2}+ \epsilon_i$$ where $$\epsilon_i \overset{iid}{\sim} N(0, \sigma^2)$$.

How do I test the null hypothesis that $$\beta_1 + 10\beta_2 = 0$$?

The variance of a sum of two random variables is given by $$\operatorname{Var}(aX+bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)+2ab\operatorname{Cov}(X,Y)$$
where $$\operatorname{Cov(⋅, ⋅)}$$ is the covariance.
In your case, substitute $$a = 1$$, $$b=10$$, $$X = \beta_1$$, and $$Y=\beta_2$$. Statistical software will typically provide a variance-covariance matrix for the coefficients of a linear regression model (although you might have to look into the documentation to figure out how to find it). The individual coefficient variances are along the diagonal, the $$(\beta_1,\beta_1)$$ and $$(\beta_2,\beta_2)$$ entries respectively. The $$(\beta_1,\beta_2)$$ entry of that matrix provides the desired covariance value.
The square root of that variance of the weighted sum of the coefficients is the estimate of its standard deviation. With a sufficiently large number of cases, if $$\beta_1+10\beta_2$$ is more than 1.96 standard deviations away from 0, then you have a significant difference at p < 0.05 in a 2-sided z-test (equivalent to a Wald test). With a smaller number of cases you could use a t-test .