6.03a - Solving linear equations
An equation says that two expressions have the same value. You will keep that balance while undoing operations, collecting unknown terms, and handling brackets and fractions. You will also turn a word problem into an equation and decide what its solution means in the original situation.
Undoing operations
Five identical notebooks and a £7 delivery charge cost £32 altogether. How much does one notebook cost?
If each notebook costs pounds, the equation is . Removing the delivery charge leaves £25 for five notebooks, so each costs £5. Algebra records those same inverse steps.
Equation and solution
An equation states that two expressions are equal. A solution is a value of the unknown that makes that statement true. For example, solves , because .
Adding or subtracting the same amount on both sides preserves equality. Multiplying or dividing both sides by the same nonzero number also preserves equality. To undo a sequence of operations, work backwards: undo the addition or subtraction before undoing the multiplication. Dividing by zero is never a valid step.
Worked example 1: reversing two operations
Question. Solve .
Plan. Subtract from both sides, then divide both sides by .
Answer. .
Check. Substitute into the original: , matching the right side.
Change only the sign of the constant on the left. The inverse operation changes too, and the solution need not be a whole number.
Worked example 2: undoing subtraction
Question. Solve .
Plan. Add to both sides before dividing by .
Answer. .
Check. . A fractional or decimal solution is allowed unless the context restricts the unknown.
The word “move” can hide what is happening. You are not simply carrying a term across an equals sign: you are doing the same operation to both sides.
Watch out
From , writing uses the wrong inverse. Subtract from both sides to remove the .
Solve, then substitute to check your result.
Unknowns on both sides
When the unknown appears on both sides, first collect its terms on one side by the same balancing operations.
For example, two hire plans cost pounds and pounds for hours. Equal cost means . Subtracting the shared leaves the extra to balance the difference in fixed charges. This is why collecting terms works.
Worked example 3: collecting the unknown terms
Question. Solve .
Plan. Subtract , then remove the constant and divide.
Answer. . In the hire-plan example, the costs are equal after five hours.
Check. The original left side is ; the right side is .
Change only the sign of the constant on the right. A negative answer is just as valid for an equation without a restrictive context.
Worked example 4: a negative solution
Question. Solve .
Plan. Collect the unknown terms in the same way.
Answer. .
Check. Left side: . Right side: . Both sides agree.
You could instead subtract and obtain a negative coefficient. That is equally valid if every operation is applied to both sides.
Sometimes all unknown terms cancel. The equation reduces to and is true for every real . But reduces to , so it has no solution. Do not divide by a zero coefficient to force a single answer.
Key idea
An ordinary linear equation with a nonzero final coefficient has one solution. Complete cancellation can instead reveal an identity or an impossible statement.
Keep the sign of the final constant in this equation.
Equations with brackets and fractions
Brackets and numerical fraction denominators change the preparation, but equality is still preserved by balancing operations.
Suppose three packs each contain counters and their total equals counters. This gives . Expand to get , then , so . Each pack has five counters, and .
For fractions, multiply every term on both sides by a common multiple of the numerical denominators. This clears the fraction bars. Keep brackets around each whole numerator while multiplying; the operation applies to all of it.
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