Solve linear equations in one variable, including equations with variables on both sides and with letter coefficients, and recognize when an equation has one solution, no solution, or infinitely many (LA A1: A-REI.B.3).

What this topic is asking

Standard A1: A-REI.B.3 asks you to solve a linear equation in one variable, including equations with the variable on both sides and with letter coefficients, and to recognize the three solution cases. On LEAP 2025 these are Type I Major Content items, and linear solving is a core no-calculator skill, so expect it in Session 1a. You both solve and (on some items) identify the number of solutions.

The properties of equality

Every solving step is one of these, applied to both sides at once.

LEAP reasoning items may show a worked solution and ask which property justifies a step, so be ready to name the move, not just perform it.

A solving routine

No solution and infinitely many solutions

When you simplify and the variable disappears, read the leftover statement:

• A false numeric statement ($5 = -2$) means the equation is never true: no solution.
• A true numeric statement ($5 = 5$) means the equation is always true: infinitely many solutions (an identity).

These appear as quick multiple-choice or inline-choice items. Keep simplifying until the variable cancels, then judge the remaining statement.

Letter coefficients

A1: A-REI.B.3 explicitly includes letter coefficients, which you handle exactly like numbers. To solve $ax + b = c$ for $x$, subtract $b$ and divide by $a$: $x = \dfrac{c - b}{a}$. This bridges directly to the literal-equations topic, where the whole equation is in letters.

How LEAP examines this topic

• Equation response. Solve for the variable and enter the exact value, including fractions.
• Multiple choice or inline choice. Identify the number of solutions, or which property justifies a step.
• Session 1a (no calculator). Linear solving is a core fluency skill tested without a calculator.

Why every step is reversible

The properties of equality work because each is reversible: if you add $3$ to both sides, subtracting $3$ returns the original equation, so the solution set never changes. This is the deep reason a check should always succeed, the steps only rewrite the same equation in a simpler form. The one operation that can break this is multiplying or dividing by an expression that might be zero, which is why the multiplication property specifies a nonzero quantity. In pure linear equations you divide only by the numeric coefficient, so this rarely bites, but it becomes important with rational equations where a denominator could vanish. Understanding reversibility also explains the special cases: an identity like $2(x + 3) = 2x + 6$ is the same expression written two ways, so every value works, while a contradiction like $x = x + 1$ asks for a number equal to one more than itself, which no value satisfies.

A worked equation with variables on both sides

Variables on both sides are the most common Session 1a format, and the move is always to gather them on one side.

It usually saves arithmetic to move the smaller variable term, so the variable you keep stays positive and you avoid a sign flip.

Try this

Q1. Solve $5(x + 2) = 3x + 16$. [2 points]

• Cue. $5x + 10 = 3x + 16 \Rightarrow 2x = 6 \Rightarrow x = 3$.

Q2. How many solutions does $4(x + 1) = 4x + 4$ have? [1 point]

• Cue. $4x + 4 = 4x + 4 \Rightarrow 4 = 4$, true: infinitely many.