Solving equations in one and two variables by keeping balance
A linear equation is one in which the variables appear only to the first power, such as 3x + 5 = 20 or 2x + 3y = 12. To solve an equation in one variable, do the same operation to both sides so the balance is kept: subtract 5 from both sides of 3x + 5 = 20 to get 3x = 15, then divide by 3 to get x = 5. Substituting the answer back into the original equation is the surest check.
A linear equation in two variables has infinitely many solutions, and when they are plotted they form a straight line. A pair of such equations can be solved together by substitution, by elimination or by drawing both lines. If the lines cross at one point there is a unique solution; if they are parallel there is none; if they coincide there are infinitely many. For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, a unique solution exists when a₁/a₂ ≠ b₁/b₂. Word problems on ages, fractions, costs and speeds are solved by naming the unknowns and turning each sentence into an equation.
Why it matters
Linear equations are the most common way to turn a real situation into something you can calculate: comparing two mobile plans, splitting a bill fairly, or working out how many tickets of each price were sold. The idea of a consistent or inconsistent pair returns with matrices and determinants in Class 12, and in linear programming, which businesses use for planning decisions.
Key terms
- Linear equation
- An equation whose variables appear only to the power one, so its graph is a straight line.
- Substitution method
- Solving a pair of equations by expressing one variable from one equation and putting it into the other.
- Elimination method
- Adding or subtracting multiples of two equations so that one variable cancels out.
- Consistent pair
- A pair of linear equations that has at least one common solution.