Linear equations in one variable: solve equations that reduce to ax + b = c, handle equations with no solution or infinitely many solutions, and interpret solutions in context.

What this skill is asking

A linear equation in one variable is any equation that, after simplifying, has the variable to the first power and reduces to the form $ax + b = c$. Solving it means isolating the variable. On the Digital SAT, the College Board (Algebra domain) also tests the two special cases (no solution and infinitely many solutions) and asks you to interpret a solution in a real-world context.

The solving routine

Every one-variable linear equation yields to the same steps.

A worked solve with fractions

Fractions are common on the SAT and are best cleared first.

No solution and infinitely many solutions

These cases are an SAT favourite because they test understanding, not just procedure. When you simplify and the variable disappears, the equation is no longer about a value of $x$; it is a statement that is either always true or never true. If you reach $7 = 7$ (or any true statement), every $x$ works, so there are infinitely many solutions. If you reach $7 = 2$ (or any false statement), no $x$ works, so there is no solution. Questions often give a parameter and ask which value forces one of these cases: set the variable coefficients equal for the lines to coincide or stay parallel, then compare constants.

Interpreting solutions in context

The SAT frequently wraps a one-variable equation in a word problem and asks what the solution means. If $C = 0.10m + 20$ models a phone plan's monthly cost $C$ in dollars for $m$ minutes, then solving $0.10m + 20 = 35$ gives $m = 150$, which means 150 minutes of calls cost $35. The algebra is the same; the extra skill is translating the situation into an equation and translating the solution back into the units of the problem. Always check that the answer is reasonable in context (a negative number of minutes, for instance, would signal an error).

A related context question gives the equation and asks you to solve for a specified variable in terms of the others, called "literal" equations or rearranging a formula. The moves are identical to solving a numeric equation: treat every letter except the target as a constant, then isolate the target. For example, to solve $P = 2\ell + 2w$ for $w$, subtract $2\ell$ from both sides to get $P - 2\ell = 2w$, then divide by $2$ to get $w = \frac{P - 2\ell}{2}$. Because you are manipulating symbols rather than numbers, there is no arithmetic to slip on, but the same rules (do the same operation to both sides, distribute carefully) apply. Recognising that a "solve for $w$" question is just the one-variable routine with letters in place of numbers keeps these from feeling harder than they are.