What is the balanced equation for the reaction of oleic acid and sodium hydroxide?

First you need the formula for the two starting ingredients. Sodium hydroxide is NaOH and Tartaric acid is HOOC--CH(OH)--CH(OH)--COOH. This can be found by looking at for example, the related link. Other searches will show that NaOH reacts with a carboxylic acid thus: -----COOH + NaOH -----> COONa +H2O in general terms.

So putting all this information together we know that 2 molecules of NaOH will be needed per molecule of tartaric acid. So the final reaction equation is

HOOC-CH(OH)-CH(OH)-COOH +2NaOH ----> NaOOC-CH(OH)-CH(OH)-COONa + 2H20