Distributive Property with Variables Calculator

Enter Expression

Multiplier (outside)

Examples: -3b 1/2a 2.5c

Parentheses expression

Use letters for variables; integers, fractions, or decimals for coefficients.

Options
Combine like terms Sort variable letters Allow fractions/decimals Show LaTeX

Download CSV Shareable Link

Result

Expanded & simplified: 6ax -15ay + 12a

Steps

3a × 2x = +6ax
3a × -5y = -15ay
3a × 4 = +12a

Share or bookmark this configuration: ?mult=3a&expr=2x-5y%2B4&allow_frac=1&sortvars=1&combine=1&show_latex=0

Formula Used

a(b + c + …) = ab + ac + …

Multiply the outside factor with each term inside parentheses, then combine like terms. Variable parts multiply by concatenation, e.g., 2a × 3x = 6ax.

Signs distribute: -k(…) flips each term's sign.
Like terms share identical variable letters and exponents.
Fractions remain exact; decimals convert to reduced fractions.

How to Use This Calculator

Enter the multiplier that sits outside the parentheses.
Enter the expression inside parentheses using integers, fractions, or decimals.
Choose options: combine terms, sort variables, fractions/decimals, LaTeX.
Press Expand & Simplify to see steps and the final result.
Export as CSV or print to PDF; share the link for revisiting.

Supported: letters a–z; integer, fractional, or decimal coefficients; plus and minus signs. Variables are concatenated on multiplication and matched exactly for like-term combining.

Example Data Table

#	Multiplier	Expression	Options	Expanded & Simplified
1	3a	2x-5y+4	combine, sort, frac	6ax - 15ay + 12a
2	-2	5m-3n+7	combine, sort	-10m + 6n - 14
3	1/2b	-x+y-2	combine, sort, frac	-1/2bx + 1/2by - b
4	2.5c	3p+2q-1	combine, frac	7.5cp + 5cq - 2.5c

Mini Glossary

Coefficient: The numerical factor multiplying a variable part, e.g., the 3 in 3x.
Variable: A letter representing an unknown quantity, such as x, y, or a.
Term: A product of a coefficient and optional variables, like -2ab or 7.
Like Terms: Terms with identical variable letters and exponents; coefficients can be added.
Distribute: Multiply the outside factor by each term inside parentheses.
Parentheses Expression: The sum or difference inside brackets to be distributed across.
Fraction: A number written a/b. The calculator reduces results to simplest form.
Decimal: A base‑10 number like 2.5. Converted to exact fractions for arithmetic.
LaTeX: Typesetting format for math. Toggle to preview steps and results.

Example: Using This Calculator

Two worked examples show the step-by-step distribution and simplification.

Example 1 — Integers

Input: Multiplier 3a, Expression (2x - 5y + 4)

3a × 2x = 6ax
3a × -5y = -15ay
3a × 4 = 12a

Result: 6ax - 15ay + 12a

Example 2 — Fractions

Input: Multiplier 1/2b, Expression (-x + y - 2)

1/2b × -x = -1/2bx
1/2b × y = 1/2by
1/2b × -2 = -b

Result: -1/2bx + 1/2by - b

Tip: Toggle “Show LaTeX” to preview typeset steps and results.

Key Rules & Identities

k(b + c + d) = kb + kc + kd
(-k)(b + c) = -kb - kc
k(b - c) = kb - kc
0·(b + c) = 0
Reverse (factoring): kb + kc = k(b + c)
Variables concatenate on multiplication and must match exactly to combine.

Like-Term Identification Quick Reference

Term A	Term B	Variable Parts	Combine?	Reason
4ax	-7xa	ax vs ax	Yes	Identical letters
3ab	5ba	ab vs ab	Yes	Order normalized
2a	-5a	a vs a	Yes	Same variable
6ab	3abc	ab vs abc	No	Different variables
7x	7x^2	x vs xx	No	Different exponents

Sign Distribution Matrix

Multiplier	Term Sign	Product Sign	Example
+	+	+	3a × 2x = +6ax
+	-	-	3a × -5y = -15ay
-	+	-	-2b × 4m = -8bm
-	-	+	-2b × -3n = +6bn

Negative outside the parentheses flips every term inside.

Sample Inputs & Expected Outputs (Validation Set)

#	Input Multiplier	Input Expression	Options	Expected Simplified Result
1	-3b	4x-5y+7	combine, sort	-12bx + 15by - 21b
2	1/2a	6m-3n+8	combine, sort, frac	3am - 3/2an + 4a
3	2.5c	-x+y-2	combine, frac	-2.5cx + 2.5cy - 5c
4	-5	3p+2q-1	combine	-15p - 10q + 5

Use these to verify consistency across CSV export or LaTeX preview.