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! Below is a real Regents problem from 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.

For more practice with linear equations, check out these resources:

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:linear-equations-graphs

https://www.ixl.com/math/grade-8/convert-a-linear-equation-in-standard-form-to-slope-intercept-form