Solve linear equations in one variable, including equations with variables on both sides and rational coefficients, and identify equations with one, none, or infinitely many solutions (A.PAR, Patterning and Algebraic Reasoning).

What this topic is asking

This Patterning and Algebraic Reasoning (A.PAR) standard is the bedrock of the whole course: solving a linear equation in one variable. The Georgia Milestones EOC tests not only getting the number but recognizing the three outcomes a linear equation can have, one solution, no solution, or infinitely many, which depends on what happens to the variable terms. These appear as numeric-entry items (solve for $x$) and as multiple-choice items asking how many solutions an equation has, often on the non-calculator section where the arithmetic is meant to be done by hand.

The basic method

A linear equation is solved by performing the same operation on both sides until the variable is alone. Work in a consistent order:

1. Distribute to remove parentheses.
2. Clear fractions by multiplying every term by the least common denominator.
3. Collect like terms: variable terms on one side, constants on the other.
4. Divide by the coefficient of the variable.

Clearing fractions

When an equation has fractions, multiplying every term by the least common denominator (LCD) removes them and avoids fraction arithmetic.

For $\frac{x}{2} + \frac{1}{3} = \frac{5}{6}$, the LCD is 6. Multiply each term by 6: $3x + 2 = 5$, so $3x = 3$ and $x = 1$. The key is to multiply every term, including those without a fraction, by the LCD.

One, none, or infinitely many solutions

The number of solutions is revealed when you simplify. The decisive moment is what happens to the variable terms.

• One solution. The variable terms do not cancel; you reach $x =$ a number. Example: $x - 6 = 7$ gives $x = 13$.
• No solution. The variable terms cancel and leave a false statement, like $6 = 5$. The equation is a contradiction; no value of $x$ works.
• Infinitely many solutions. The variable terms cancel and leave a true statement, like $6 = 6$. The equation is an identity; every value of $x$ works.

How the Milestones examines this topic

• Numeric entry. Solve a multi-step equation, possibly with fractions or distribution, and type the value.
• Multiple choice. Identify the number of solutions, with "one," "none," and "infinitely many" as options.
• Constructed response. Solve and justify, or set up an equation from a context and solve it.

Why canceling variables tells you the solution count

The three outcomes are not arbitrary; they come from the geometry behind the algebra. A linear equation in one variable can be thought of as asking where two lines $y =$ (left side) and $y =$ (right side) meet. If the lines have different slopes, they cross once, which is the one-solution case. If they are the same line (same slope and intercept), they overlap everywhere, which is infinitely many solutions, the identity. If they are parallel but distinct (same slope, different intercept), they never meet, which is no solution, the contradiction. That is why watching the variable terms is so informative: identical variable terms mean equal slopes, and then the constants decide between "same line" and "parallel."

Equations from context

Many EOC items hide a linear equation inside a sentence. "A gym charges a $25 fee plus$15 per month; after how many months is the total $115?" becomes$25 + 15m = 115$, so$15m = 90$and$m = 6$ months. The skill is translating the words into the equation, then solving by the same steps. Reading the fixed amount as the constant and the repeated amount as the coefficient of the variable, exactly the structure idea from the expressions strand, makes the translation reliable, and stating the units in the answer (6 months) is part of the modeling credit.

Try this

Q1. Solve $5x - 8 = 3x + 10$. [1 point]

• Cue. $2x = 18$, so $x = 9$.

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

• Cue. $4x + 4 = 4x + 4$ gives $4 = 4$, true, so infinitely many.