Solve linear equations in one variable, including those with variables on both sides and with the distributive property, and recognize no-solution and identity cases (Ohio A-REI.3, A-REI.1).

What this topic is asking

Ohio standard A-REI.3 asks you to solve linear equations in one variable, and A-REI.1 asks you to justify the steps. That includes equations with parentheses, fractions, and variables on both sides, plus the special cases of no solution and infinitely many solutions. This is a bedrock skill in the Expressions and Equations reporting category, and it appears on Part 1 where no calculator helps.

The solving routine

The reliable order of operations for solving is the reverse of building an expression.

Clearing fractions

When an equation has fractions, multiply every term by the least common denominator first; this avoids fraction arithmetic later.

For $\dfrac{x}{2} + \dfrac{1}{3} = \dfrac{5}{6}$, the LCD is $6$. Multiplying every term by $6$ gives $3x + 2 = 5$, so $3x = 3$ and $x = 1$.

No solution and infinitely many

Most linear equations have exactly one solution, but two special cases arise when the variable cancels.

The key is to finish the algebra: if you ever reach a statement with no variable, judge whether it is true or false to decide between infinitely many and none.

How Ohio examines this topic

• Equation response. Type the single solution value.
• Multiple choice. Decide how many solutions an equation has, or pick the correct value.
• Drag and drop. Order the solving steps, or match an equation to "one / none / infinitely many."

Because Part 1 bans the calculator, the arithmetic, especially distributing a negative and combining signed terms, must be clean.

Why every step preserves the solution

A-REI.1 is really asking "why are these steps allowed?" The answer is that each move, adding the same amount to both sides, multiplying both sides by the same nonzero number, distributing, applies an operation equally to both sides, so any value that made the original equation true still makes the new one true, and vice versa. That is why solving never changes the solution set: you are rewriting the same balance in a simpler form. The one move to watch is multiplying or dividing by a variable expression that could be zero, which is why for linear equations we only ever multiply by known nonzero numbers (like a common denominator). Understanding this justification is what lets you trust a chain of steps and explain it on a reasoning item.

Checking, and what a check catches

Substituting your answer back into the original equation is the fastest insurance on a no-calculator part. A check catches a distribution slip or a sign error immediately: if the two sides do not match, you know to retrace. It is especially worth doing when you distributed a negative, like $-(x - 3) = -x + 3$, since dropping that sign flip is the single most common mistake in this topic. Build the habit of a ten-second check on every equation-response item.

Try this

Q1. Solve $4x - 7 = 2x + 9$. [2 points]

• Cue. $2x = 16$, so $x = 8$.

Q2. How many solutions does $3(x + 1) = 3x + 3$ have? [1 point]

• Cue. $3x + 3 = 3x + 3$ gives $3 = 3$ (true), so infinitely many.