Equivalent equations, rational equations and word problems
Learn what equivalent equations and an extraneous root are, solve a rational equation with a check, and form equations for motion and work problems.
- Ukraine: grade 8
- Russia: grade 8
Guide
In the Ukrainian programme this is Grade 8 algebra: §1 item 7 "Equivalent equations. Rational equations" and §3 item 23 "Rational equations as models of real situations".
Two equations are equivalent if they have the same roots (or both have none). Moving a term to the other side with a changed sign, or multiplying both sides by a non-zero number, keeps the roots.
A rational equation has fractions with the unknown in the bottom. Values that make a bottom 0 are not allowed. A fraction is 0 when its top is 0 and its bottom is not. A root that makes a bottom 0 is extraneous and is dropped.
In motion and work problems name the unknown x, form the equation, solve it and check that the root fits the story. Time is distance divided by speed; in one hour 1x of a job is done.
On the board the fractions show the times or the parts of the job. Solve the equation on paper and type the answer.