Lesson 5.9

Standard Form

Sometimes we don't want to choose sides. Standard Form puts and on the same team. It's clean, formal, and powerful for systems.

Introduction

Slope-Intercept form () is great for graphing, but it can get messy with fractions. Standard Form cleans everything up. No fractions, no decimals, just integers.

Past Knowledge

Lesson 5.7. You know how to work with . Now let's rearrange it.

Today's Goal

Write linear equations in the form .

Future Success

This is essential for solving Systems of Equations (Elimination Method) in Chapter 14.

Key Concepts

The Rules of Standard Form

Rule 1: Ax is Positive

The coefficient of (A) must be . If it's negative, flip all signs.

Rule 2: No Fractions

A, B, and C must be integers. Multiply the whole equation to clear denominators.

Rule 3: Greatest Common Factor

Simplify if possible. should be divided by 2 to get .

Worked Examples

Example 1: Rearranging

Basic

Convert to Standard Form.

Step 1: Move x

Subtract from both sides.

-2x + y = 5

Step 2: Fix Negative A

A is -2. Multiply everything by -1.

Verification

Both equations describe the exact same line.

Example 2: Clearing Fractions

Intermediate

Convert to Standard Form.

Step 1: Move x

Step 2: Clear Fraction

Multiply EVERYTHING by 3.

Step 3: Fix Signs

Multiply by -1.

Example 3: Horizontal Lines

Advanced

Convert .

Analysis

Standard Form is .

Here, there is no , so . . .

0x + 1y = 5

Wait, it's already in Standard Form! (Technically ).

Common Pitfalls

Forgetting C

When multiplying by a common denominator (e.g., 3), students often forget to multiply the constant term on the right side (). Distribute to EVERY term.

Leaving Negative A

is technically valid algebraically, but formal "Standard Form" requires . Always flip the signs as the last step.

Real-Life Applications

Shopping Budgets: You have $50 to spend. T-shirts cost $10 () and Hats cost $5 ().

Equation: .

This is naturally in Standard Form. Converting it to is extra work if you just want to know combinations like "5 shirts, 0 hats" or "0 shirts, 10 hats".