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Copy of V4 Copy of Essential Strategies for Mastering the Regent's Exam Study Guide

Preparing for the Regent's Exam can feel overwhelming. The pressure to perform well and the vast amount of material to cover often leave students unsure where to begin. This guide offers clear, practical strategies to help you navigate the Regent's Exam Study Guide effectively and boost your confidence for test day.


The 6 Topics You Need to Know


The Algebra 1 Regents is passable — and with the right practice, very passable. This guide covers six major topics that appear on every exam, with real problems from past Regents tests. What makes this study guide a cut above the rest: you try each problem first in the scratchpad, then reveal the step-by-step solution. That's how you actually learn, not just review.


  1. Linear Equations


The most tested topic on the Regents. Get this right and you've already well on your way in points.


Click here for some tips on linear equations

Linear equations show up everywhere — solving for a variable, rearranging formulas, writing inequalities from word problems. The reliable process: isolate the variable by undoing operations in reverse order. Undo addition and subtraction first, then multiplication and division.


What to Know Cold:

  • Slope-intercept form: y = mx + b (m = slope, b = y-intercept)

  • Slope from two points: m = (y₂ − y₁) / (x₂ − x₁)

  • Rearranging a formula: treat every letter except your target as a number, and isolate it using inverse operations.


⚠The sign-flip rule: when you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. So −2x > 6 becomes x < −3. This costs students points on every single exam.

✓Partial credit is real. A correct setup with an arithmetic error still earns most of the points. Always write out your steps — never just write the answer.



Worked example


Problem: Solve for x: 3x + 5 = 20



Goal: Get numbers on one side and x on another.

  1. +5 is on x's side — subtract it from both sides.

  2. 3 is still on x's side — divide both sides by 3.

  3. x is alone. All done!









Try it yourself→

Try it yourself!


Real Regents Problem, January 2024


Problem: Solve for x: (2/3)(3 − 2x) = 3/4.



Let's get started.

  1. Type your next step below the problem

  2. Press enter to add a new line of math 

  3. Our AI checks your work as you go 🟢means you're on the right track, ❌means something's off.

  4. Made a mistake? Use Ctl+Z to undo a line. Right click to see other shortcuts.




Click here to see the answer

STEP-BY-STEP SOLUTION

  1. Distribute 2/3 across the parentheses. (2/3)(3) − (2/3)(2x) = 3/4 2 − 4x/3 = 3/4

  2. Multiply every term on both sides by 12 to eliminate fractions. 12(2) − 12(4x/3) = 12(3/4) 24 − 16x = 9

  3. Subtract 24 from both sides. −16x = 9 − 24 −16x = −15

  4. Divide both sides by −16. x = −15 / −16 x = 15/16

Answer:x = 15/16

✗ COMMON TRAP

Forgetting to distribute the 2/3 to both terms inside the parentheses — only multiplying 2/3 by 3 and leaving the −2x untouched.

✓ KEY IDEA

Multiplying both sides by 12 (the LCM of 3 and 4) is the fastest way to clear all fractions at once. Once the fractions are gone the rest is straightforward.



  1. Solving Systems of Equations

Learn to spot the two equations hiding inside any real-world setup.


Every system gives you two facts about two unknowns. Write those facts as equations, then solve. The Regents loves coin problems, pricing problems, and age problems — they look different but use the exact same method.


The trick is about setting each unknown as one half of an equation, then using one of the two methods below to find the answer of one equation by using the other!


Click here for some tips on systems of equations

Here are the two ways to solve a system of equations


Substitution: solve one equation for a variable, substitute into the other.


Elimination: add or subtract equations to cancel a variable. Multiply one first if needed.


Use whichever feels more natural — the Regents doesn't care which method you use.


⚠Label before you solve. Write "let n = nickels, let q = quarters" before writing any equations. Students who skip this often mix up variables midway through and lose points on an otherwise correct solution.



Worked example


Problem: Solve the system of equations:

y = 2x

x + y = 12


Substitution Method

Goal: When you have two equations with two unknowns, you need to use one equation to eliminate a variable from the other — leaving you with one equation and one unknown, which you already know how to solve.




  1. Y is already isolated in equation (1), so we know y = 2x. Anywhere we see y in equation (2), we can replace it with 2x — now equation (2) only has x in it, which we can solve.

  2. Now that we know x = 4, we can substitute it back into either equation. We now have only one unknown left (y), so we can isolate it and solve.



OR


Elimination Method


Goal: Find a variable that appears with the same coefficient in both equations. Subtracting one equation from the other will then cancel that variable out — leaving you with one equation and one unknown.




  1. Both equations have y with a coefficient of 1, so we can subtract the top equation from the bottom equation directly.

  2. The y terms cancel, leaving only x. Solve for x as normal.

  3. Now that we know x = 4, substitute it back into either equation to find y.






Try it yourself!


Real Regents Problem, January 2024


Problem: Jim had a bag of coins. The number of nickels, n, and quarters, q, totaled 28 coins. Their combined value was $4.00. Write a system of equations that models this situation. Algebraically determine both n and q.



Let's get started.

  1. Type your next step below the problem, or try clicking a number and dragging it to the other side of the "=".

  2. Press enter to add a new line of math 

  3. Our AI checks your work as you go 🟢means you're on the right track, ❌means something's off.

  4. Made a mistake? Use Ctl+Z to undo a line. Right click to see other shortcuts.



Click here to see the answer

STEP-BY-STEP SOLUTION

  1. Write the system:

    n + q = 28


    0.05n + 0.25q = 4.00

  2. Multiply the first equation by 0.05:

    0.05n + 0.05q = 1.40

  3. Subtract from the second equation to eliminate n:

    0.20q = 2.60 → q = 13

  4. Substitute back:

    n + 13 = 28 → n = 15

  5. Check: 15(0.05) + 13(0.25) = 0.75 + 3.25 = $4.00 ✓

15 nickels and 13 quarters. Always verify — one check line can catch a careless error.

✗ COMMON TRAP

Writing 5n + 25q = 4 — mixing cents and dollars. If you use cents on the left, the right side must be 400, not 4.

✓ THE FIX

Choose one unit and stick to it. All dollars: 0.05n + 0.25q = 4. All cents: 5n + 25q = 400. Both work.



  1. Reading and interpreting functions

Function problems on the Regents are mostly about reading carefully and applying slope-intercept form correctly.


The Regents tests functions a few consistent ways: identifying whether a relation is a function, interpreting the parts of a linear equation in context, and reading transformations of parabolas. Once you know what to look for, these are reliable points.


Click here for tips on reading and interpreting functions

The essentials

  • A relation is not a function if any x-value appears more than once with a different y-value.


  • In y = mx + b: m is the rate of change per unit; b is the starting value when x = 0.


  • Transformations of y = x²:


  •   (x + 3)² shifts the graph left 3 (sign is opposite the direction

  •   (x − 3)² shifts it right 3

  •    x² − 2 shifts it down 2


⚠Left/right shifts feel backwards. (x + 3)² moves the vertex left to x = −3. Quick check: set the inside equal to zero — x + 3 = 0 gives x = −3. That's where the new vertex sits.



Worked Example


Problem: Marcus is saving money. He starts with $40 and saves $15 every week. The total amount he has saved after x weeks is modeled by y = 40 + 15x.

(a) How much money did Marcus start with? (b) How much does he save each week? (c) How much will he have after 6 weeks?




Try it yourself!


Real Regents Problem, January 2024



Problem:The Speedy Jet Ski Rental Company models its total cost as y = 30 + 40x, where x is hours rented and y is total cost in dollars.

(a) What is the insurance fee?  (b) What is the hourly rental rate?  (c) What is the total cost for a 3-hour rental?



Let's get started.

  1. Type your next step below the problem, or try clicking on a number and dragging it to the other side of the "=".

  2. Press enter to add a new line of math 

  3. Our AI checks your work as you go 🟢means you're on the right track, ❌means something's off.

  4. Made a mistake? Use Ctl+Z to undo a line. Right click to see other shortcuts.



Click here to see the answer

STEP-BY-STEP SOLUTION

  1. Match to slope-intercept form y = mx + b. Rewriting: y = 40x + 30, so m = 40 and b = 30.

  2. (a) Insurance fee = b = $30 — the flat fee before any hours.

  3. (b) Hourly rental rate = m = $40 per hour.

  4. (c) Substitute x = 3:

    y = 30 + 40 * 3 = 30 + 120 = $150

Insurance fee: $30. Hourly rate: $40/hr. 3-hour rental: $150.

✗ COMMON TRAP

Swapping slope and intercept — calling $40 the fee and $30 the rate. The constant is always the flat fee; the coefficient of x is always the per-unit rate.

✓ QUICK CHECK

What does the company charge for 0 hours? That's the flat fee — and it's the constant, since 40 × 0 = 0.




  1. Quadratics

Quadratics show up in both Part I and the longer constructed-response questions. Worth knowing cold.


Click here for some tips about quadratics

Quadratic questions come in a few forms: factoring a polynomial, solving by factoring or the quadratic formula, interpreting a parabola from a graph, and working with real-world height problems. The golf ball problem is practically a Regents staple at this point.

  • Your toolkit

  • Standard form: ax² + bx + c = 0


  • Axis of symmetry (x-coordinate of vertex): x = −b / (2a)


  • Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)


  • "Factor completely" = pull out the GCF first, then factor the remaining trinomial.


✓Height problems always follow the same pattern. When does it hit the ground? Set the expression equal to 0 and solve. Max height? Find t = −b / (2a), substitute back in. What does the constant represent? It's the height at t = 0.



Try it yourself!:


Warm-up problem (Level 1)


Problem: Solve by factoring: x² − 5x + 6 = 0




Exam-level problem — Regents-style


Problem: Laura hits a golf ball from the ground. Its height in feet is modeled by −16t² + 48t, where t is time in seconds.


(a) When does the ball hit the ground?  (b) What is the maximum height?  (c) At what time does it reach maximum height?


Type the equation in the box next to the word "let" to get started



Let's get started.

  1. Type your next step below the problem, or try clicking on a number and dragging it to the other side of the "=".

  2. Press enter to add a new line of math 

  3. Our AI checks your work as you go 🟢means you're on the right track, ❌means something's off.

  4. Made a mistake? Use Ctl+Z to undo a line. Right click to see other shortcuts.



Click here to see the answer

  1. (a) Set the expression equal to zero and factor:

    −16t² + 48t = 0


    −16t(t − 3) = 0


    t = 0 (when hit) or t = 3 seconds (when it lands)

  2. (b) and (c) Axis of symmetry gives the time of max height:

    t = −48 / (2 × −16) = 48 / 32 = 1.5 seconds

  3. Substitute t = 1.5 into the height expression:

    −16 × (1.5)² + 48 × 1.5 = −16 × 2.25 + 72 = −36 + 72 = 36 feet

Lands at t = 3 sec. Max height: 36 feet at t = 1.5 sec.

✗ COMMON TRAP

Reporting t = 3 as the time of max height. The zeros tell you when it's on the ground — the vertex tells you the max.

✓ THE RULE

Zero of the expression = on the ground. Vertex = highest point. One question about each.

  1. Exponents and Compound Interest

Once you know a handful of rules, exponent questions become fast, reliable points. Compound interest is on almost every exam.


The Regents tests exponent rules — multiplying powers, raising a power to a power — and exponential growth, which is what happens when something multiplies by a constant rate each period. Compound interest is the classic example.


Click here for some tips on exponents and compound interest

Rules worth knowing

  • xᵃ × xᵇ = x^(a+b)  (multiply → add exponents)

  • (xᵃ)ᵇ = x^(a×b)  (power of a power → multiply exponents)

  • (xy)ⁿ = xⁿ × yⁿ  (distribute the exponent)

  • Compound interest: A = P × (1 + r)^t

  • P = starting amount, r = rate as a decimal, t = time


⚠3% as a decimal is 0.03 — not 0.3. Move the decimal point two places left. Then the growth factor is 1 + 0.03 = 1.03. Confusing 3% with 30% is one of the most common errors on every Regents.

Worked example


Problem: A population of bacteria doubles every hour starting from 500. Which equation gives the population y after x hours?


(1) y = 500 + 2x    (2) y = 500 × 2^x    (3) y = 2 × 500^x


Type the equation in the box next to the word "let" to get started



Try it yourself!:


Real Regents Problem, January 2024


Problem: Joe deposits $4,000 into a CD earning 3% interest compounded annually. Which equation gives the value y after x years? (1) y = 4000 + 0.3x (2) y = 4000 + 0.03x (3) y = 4000 × (1.3)^x (4) y = 4000 × (1.03)^x


Type the equation in the box next to the word "let" to get started


Let's get started.

  1. Type your problem where the "..." is.

  2. Press enter to add a new line of math 

  3. Our AI checks your work as you go 🟢means you're on the right track, ❌means something's off.

  4. Made a mistake? Use Ctl+Z to undo a line. Right click to see other shortcuts.





Click here to see the answer

  1. "Compounded annually" = exponential. Eliminate (1) and (2).

  2. Convert rate: 3 / 100 = 0.03.

  3. Growth factor: 1 + 0.03 = 1.03.

  4. Equation: y = 4000 × (1.03)^x.

Answer: (4). Choice (3) uses 1.3 — that would be 30% growth, not 3%.

✗ COMMON TRAP

Choosing 1.3 as the base — treating 3% as 0.3. Always divide the percentage by 100 first.

✓ SANITY CHECK

At 3% interest, your money grows very slowly. A growth factor of 1.3 would mean 30% per year — that's not a normal bank rate.


  1. Statistics

No algebra required. These are some of the most approachable questions on the exam — just division and careful reading.


Two-way table questions give you a grid of data and ask for a percentage. The only trick is finding the right denominator. The question tells you which group to look at — that group's total goes on the bottom of your fraction.


Click here for some tips on statistics

Three steps — every time

1. Re-read the question and find the group being asked about. Underline it.


2. Add up the total for that group — that's your denominator (the bottom).


3. Find the specific part being asked about — that's your numerator (the top).



Then: percentage = (numerator / denominator) × 100

✓Before you touch the calculator: write out "part = ___, whole = ___" and fill in both numbers from the table. This one habit prevents the most common error on these questions.


Worked example


Problem: In a class of 30 students, 18 prefer math and 12 prefer English. Of the students who prefer math, 10 are in 9th grade and 8 are in 10th grade.

What percentage of the students who prefer math are in 10th grade?







Try it yourself!:


Real Regents Problem, January 2024


Problem: Mrs. Smith surveyed students about their favorite ice cream. Results: Juniors — chocolate: 42, vanilla: 27, twist: 45. Seniors — chocolate: 67, vanilla: 42, twist: 21.

Of the students who preferred chocolate, approximately what percentage were seniors?

(1) 27.5% (2) 44.7% (3) 51.5% (4) 61.5%


Let's get started.

  1. Type math lines where is says "Click here to write math"

  2. Press Alt/Option+Enter after typing a line of math to calculate that line, just like a calculator

  3. Our AI checks your work as you go 🟢means you're on the right track, ❌means something's off.

  4. Made a mistake? Use Ctl+Z to undo a line. Right click to see other shortcuts.



Click here to see the answer

Solution: Step 1: Total who chose chocolate: 42 + 67 = 109. Step 2: Seniors who chose chocolate: 67. Step 3: Calculate: 67 / 109 × 100 ≈ 61.5%

Answer: (4) 61.5%. Denominator = 109, not 244.

✗ The trap: Using 244 (all students) gives 27.5% — answer (1). It's designed to catch students who don't identify the right group. ✓ The habit: "Part = 67, whole = 109." Writing this out before calculating takes five seconds and prevents this mistake every time.


Catch your algebra mistakes before they cost you

The scratchpad in this guide is powered by Moment of Math — a free Chrome extension that checks your algebra work line by line as you write it, on any problem. Try it on your homework tonight.




 
 
 

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