A general strategy, and equations with no or every solution
Solve 2y + 6 = 2(y + 3) and the y disappears, leaving 6 = 6. Every number is a solution.
Five steps that work on any linear equation
By now you have every tool needed for linear equations in one variable. OpenStax gathers them into a general strategy. Step 1: simplify each side as much as possible, using the distributive property to clear parentheses and combining like terms. Step 2: collect all the variable terms on one side. Step 3: collect all the constant terms on the other side. Step 4: make the coefficient of the variable 1, by dividing or multiplying. Step 5: check the solution in the original equation. Not every equation needs every step, but going through them in order keeps long problems under control.
An example from the textbook: −6(x + 3) = 24. Distribute the −6 to both terms inside the parentheses, which gives −6x − 18 = 24. There is only one variable term, so step 2 is done. Add 18 to both sides to get −6x = 42. Divide by −6 to get x = −7. Check: −6 × (−7 + 3) = −6 × (−4) = 24. The most common error here is distributing the −6 to the x but not to the 3, or dropping its sign. Write the multiplication out, −6 · x + (−6) · 3, if it helps.
Three kinds of equation
Most equations are true for one value and false for all others, like 7x + 8 = −13, which is true only when x = −3. OpenStax calls these conditional equations. Now try 2y + 6 = 2(y + 3). Distribute on the right: 2y + 6 = 2y + 6. Subtract 2y from both sides and you are left with 6 = 6. The variable has vanished and the statement that remains is true. That means every value of y works, and the equation is called an identity. Its solution is all real numbers.
The third kind goes the other way. Solve 5z = 5z − 1 by subtracting 5z from both sides, and you get 0 = −1. The variable is gone again, but this time the statement left over is false, and no value of z can make it true. Such an equation is a contradiction, and it has no solution. So when the variable disappears while you are solving, look at what is left: a true statement means every number, a false one means no number. Do not write x = 0 in either case.
| Kind | What is left after solving | Solutions |
|---|---|---|
| Conditional | x = a number | exactly one |
| Identity | a true statement such as 6 = 6 | all real numbers |
| Contradiction | a false statement such as 0 = −1 | none |
The graph lab shows why. For ax + b = cx + d, the two sides are lines with slopes a and c. If a and c differ, the lines cross once: a conditional equation. If a equals c and b equals d, the lines are the same line, drawn on top of each other: an identity. If a equals c but b differs from d, the lines are parallel and never meet: a contradiction. Set a = 2, b = 6, c = 2, d = 6 and watch the orange line hide the blue one.
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- Solve −6(x + 3) = 24. x = ?
- Solve 7x + 8 = −13. x = ?
- Solve 4(n − 3) = 8. n = ?
- Solve 3(p + 2) + 2p − 5 = 2p + 10. p = ?
- Solve 2(y − 3) = 9 − y. y = ?
- Classify 2y + 6 = 2(y + 3).
- Identity: all real numbers
- Contradiction: no solution
- Conditional: y = 0
- Conditional: y = 3
- Classify 5z = 5z − 1.
- Identity: all real numbers
- Contradiction: no solution
- Conditional: z = 0
- Conditional: z = 1
- Classify 3(x + 1) = 3x + 4.
- Identity: all real numbers
- Contradiction: no solution
- Conditional: x = 1
- Conditional: x = 4
- Classify 4(2a − 1) = 8a − 4.
- Identity: all real numbers
- Contradiction: no solution
- Conditional: a = 1
- Conditional: a = 0
- Put the steps of the general strategy in order.
- Simplify, collect variables, collect constants, make the coefficient 1, check.
- Distribute a number to every term in the parentheses, with its sign.
- Variable gone and a true statement left: identity, all real numbers.
- Variable gone and a false statement left: contradiction, no solution.