Equivalent equations and exact solutions
Solve by transformations that preserve the solution set. Keep signs, brackets and whole numerators intact, and verify the answer in the original equation
- Equality and equivalence An equality says two quantities have the same value. Equivalent equations have the same solution set. Apply the same addition or subtraction to both sides; use a non-zero number for multiplication or division
- Unknowns on both sides Subtract a chosen unknown term from both sides, then collect and isolate the remaining unknown
- Cancellation cases A false constant equality means no solution. A true identity means every real value allowed by the original equation is a solution
- Brackets and fractions Distribute to every bracket term. When clearing a constant denominator, multiply every term on each side. A simple bracket may instead be isolated by division first
- Exactness and checking Keep exact fractions unless rounding is requested. Substitute into the original equation and compare both sides
Worked example
Solve and verify the result
- \(\displaystyle -6x+8=x+1\)Distribute the negative factor to both terms
- \(\displaystyle 8=7x+1\)Add 6x to both sides
- \(\displaystyle 7=7x\)Subtract 1 from both sides
- \(\displaystyle x=1\)Divide both sides by 7
- \(\displaystyle -2(3(1)-4)=2\)Evaluate the original left side
- \(\displaystyle 1+1=2\)The original right side matches
Common mistakes
Difficulty levels
Questions about this topic
Why are inverse operations used in reverse order?+
Undoing the outer operation first often gives a short route to the unknown. Other equivalent operations are also valid if they are applied to the whole of both sides
Why must a bracket factor multiply every term?+
The distributive law says a factor outside a bracket multiplies each term inside it
How can a solution be checked?+
Substitute the value into the original equation and calculate both sides. They must have the same value
Can a linear equation have no solution?+
An equation built from linear expressions can reduce to a false equality such as 2 = 7. It then has no solution. If it reduces to an identity, every allowed value works
Can I divide by an expression containing x?+
That can exclude values where the expression is zero. Here use non-zero numerical divisors and subtraction of unknown terms, so no possible solution is silently lost
Must I always undo the constant first?+
No. Different valid routes work. Whatever operation you choose must act on the whole of both sides
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 8: Linear equations and inequalities