Solving linear equations
A linear equation has a variable to the first power only. Solving it means finding the value that makes both sides equal, by doing the same operation to both sides until the variable stands alone.
Equations and solutions
An equation states that two expressions are equal. A solution is a value of the variable that makes the statement true: x = 4 solves \(2x + 3 = 11\) because \(2(4) + 3 = 11\). A linear equation in one variable can be written as \(ax + b = 0\) with \(a \neq 0\), and it has exactly one solution.
The balance principle
Think of an equation as a balanced scale. Adding, subtracting, multiplying, or dividing both sides by the same quantity (except dividing by zero) keeps it balanced and produces an equivalent equation with the same solutions. The goal is to isolate the variable on one side.
Two-step equations
To solve \(2x + 3 = 11\), undo operations in reverse order: subtract 3 from both sides to get \(2x = 8\), then divide by 2 to get \(x = 4\). Always check by substituting back into the original equation.
Variables on both sides
Collect variable terms on one side and constants on the other. For \(5x - 7 = 2x + 8\), subtract 2x to get \(3x - 7 = 8\), add 7 to get \(3x = 15\), and divide to get \(x = 5\). Check: \(5(5) - 7 = 18\) and \(2(5) + 8 = 18\).
Brackets and fractions
Distribute first: \(3(x - 2) = 2(x + 1)\) becomes \(3x - 6 = 2x + 2\), so \(x = 8\). To clear fractions, multiply every term by the least common denominator: \(\frac{x}{3} + \frac{x}{4} = 7\) becomes \(4x + 3x = 84\), so \(x = 12\).
No solution or every number
Sometimes the variable disappears. If \(2(x + 3) = 2x + 5\), simplifying gives \(6 = 5\), which is false, so there is no solution. If \(2(x + 3) = 2x + 6\), you get \(6 = 6\), true for every x: an identity with infinitely many solutions.
Literal equations and formulas
The same steps rearrange formulas. To solve \(P = 2l + 2w\) for w, subtract 2l and divide by 2: \(w = \frac{P - 2l}{2}\). To convert Celsius to Fahrenheit and back, \(F = \tfrac{9}{5}C + 32\) rearranges to \(C = \tfrac{5}{9}(F - 32)\), so 98.6°F is 37°C.
Worked example
A taxi charges a $3.50 flag fee plus $2.25 per mile. How far can you go for $26? Write \(3.50 + 2.25m = 26\). Subtract 3.50 to get \(2.25m = 22.50\), then divide: \(m = 10\) miles. Check: \(3.50 + 2.25 \times 10 = 26\).
Key takeaways
A solution makes an equation true. Doing the same operation to both sides keeps an equation balanced. Undo operations in reverse order, collect variable terms on one side, distribute brackets first, and clear fractions by multiplying by the least common denominator. If the variable cancels, a false statement means no solution and a true one means every number is a solution. The same steps rearrange formulas such as P = 2l + 2w and the temperature conversion. Always check by substituting back.
Flashcard
Statement that two expressions are equal
Check your understanding
Solve \(5x - 7 = 2x + 8\).