NY Regents Algebra I: a complete guide to expressions and equations on the exam

What this strand demands

The expressions-and-equations strand is the algebraic core of NY Regents Algebra I, and it supplies a large share of the credits on every paper. The exam wants fluent skills (solve, factor, rearrange), purposeful rewriting (reveal a zero, a vertex, a rate), and clear modeling (build an equation or system from a context and interpret the answer). This guide ties together the dot-point pages, each with its own practice: interpreting and rewriting expressions, polynomial operations and factoring, linear equations and inequalities, systems of equations and inequalities, and solving quadratic equations.

Reading and rewriting expressions

An expression's structure carries meaning. Terms are separated by $+$ and $-$; factors are multiplied within a term; the coefficient is the numeric factor. In a context, each piece has a real interpretation: in $C = 30 + 0.10m$, the $30$ is a fixed fee and the $0.10$ is a per-minute rate.

Rewriting exposes hidden features. The same quadratic appears three ways:

For exponential expressions, writing the base as $1 + r$ or $1 - r$ reveals the percent change: $800(1.04)^t$ grows 4% per year because $1.04 = 1 + 0.04$.

Polynomial arithmetic and factoring

Add and subtract by combining like terms, distributing any subtraction sign across every term. Multiply with the distributive property. Polynomials are closed under these operations.

Factoring follows a fixed order: greatest common factor, then difference of two squares $a^2 - b^2 = (a - b)(a + b)$, then trinomial factoring. For $x^2 + bx + c$, find two numbers with product $c$ and sum $b$. For $ax^2 + bx + c$ with $a \neq 1$, use the AC method: find two numbers multiplying to $ac$ and adding to $b$, split the middle term, and factor by grouping.

Linear equations, inequalities, and literal equations

Solve linear equations by undoing operations in reverse, gathering variables on one side. Inequalities use the same moves with one rule: flip the sign when multiplying or dividing by a negative. Graph the solution on a number line with an open circle for strict inequalities and a closed circle otherwise.

A literal equation is rearranged like a numerical one, treating the target variable as the unknown. To solve $A = \frac{1}{2}bh$ for $h$, multiply by 2 and divide by $b$: $h = \frac{2A}{b}$.

Systems of equations and inequalities

A system's solution satisfies every equation at once. Use substitution when a variable is isolated, elimination when adding cancels a variable (scale first if needed), or graphing to read the intersection. A linear system has one solution, no solution (parallel lines), or infinitely many (the same line).

A linear-quadratic system is solved by substituting the line into the parabola, giving a quadratic with up to two solutions. A system of inequalities is solved by shading each half-plane; the overlap is the solution region.

Solving quadratics three ways

Always set the equation to $ax^2 + bx + c = 0$ first. Factor and use the zero-product property when it factors cleanly. Complete the square when asked or to find the vertex. Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ when factoring fails, simplifying any radical (so $\sqrt{72} = 6\sqrt{2}$). The discriminant $b^2 - 4ac$ gives the number of real roots. In a context, reject solutions that make no sense, such as a negative time or length.

How this strand is examined

• Part I (2 credits, no work shown). Quick solving, factoring, reading vertex form, counting system solutions. Substitute to check before bubbling.
• Part II (2 credits). A short factor, a literal-equation rearrangement, or a one-step inequality with a number-line graph. Show the key step.
• Part III and IV (4 to 6 credits). A systems word problem, an extended quadratic model, or a multi-step inequality. Define variables, show every step, and interpret in context.

Check your knowledge

Work these under exam conditions, showing the method as you would for credit.

1. Subtract $2x^2 - x + 3$ from $5x^2 + 4x - 1$. (2 credits)
2. Factor completely: $3x^2 - 12$. (2 credits)
3. Solve $4 - 2(x - 1) = 3x + 1$. (2 credits)
4. Solve $-5x + 3 \geq 18$ and describe the number-line graph. (2 credits)
5. Solve the system $y = x - 2$ and $2x + y = 10$. (2 credits)
6. Solve $x^2 - 2x - 15 = 0$ by factoring. (2 credits)
7. Solve $x^2 + 4x - 6 = 0$ by the quadratic formula in simplest radical form. (4 credits)
8. A store sells pens at $1.50 and notebooks at$3. A customer buys 10 items for $21. Write and solve a system. (4 credits)