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Equation Balance & Intro Algebra Lab

See equations as a balance scale. Whatever you do to one side, you must do to the other. Solve one-step, two-step, and multi-step linear equations with step-by-step inverse operations and substitution verification.

Guided Experiment: Inverse Operations

What inverse operations are needed to isolate the variable in one-step and two-step equations? Does the order of operations matter?

Write your hypothesis in the Lab Report panel, then click Next.

Balance Scale

xx3337777777Left sideRight sideBalanced!
Variable (x)Positive constantNegative value

Controls

Left side (ax + b)
=
Right side (cx + d)

Solution Steps

1Start with the equation
2x+3=72x + 3 = 7
2Subtract 3 from both sides−3-3
2x=42x = 4
3Divide both sides by 2÷2\div 2
x=2x = 2
Solution
x=2x = 2
Verification (substitute back)
2(2)+3=7=?7✓2(2) + 3 = 7 \stackrel{?}{=} 7 \checkmark

Data Table

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Reference Guide

The Balance Principle

An equation is like a balance scale. Both sides must stay equal. Any operation applied to one side must also be applied to the other.

x+5=12  ⟹  x+5−5=12−5  ⟹  x=7x + 5 = 12 \implies x + 5 - 5 = 12 - 5 \implies x = 7

This principle is the foundation of all equation solving.

Inverse Operations

To isolate the variable, use the inverse (opposite) operation to undo each step.

2x−3=7→+32x=10→÷2x=52x - 3 = 7 \xrightarrow{+3} 2x = 10 \xrightarrow{\div 2} x = 5

Addition undoes subtraction. Multiplication undoes division. Work from the outside in.

Variables on Both Sides

When the variable appears on both sides, first collect all variable terms on one side and all constants on the other.

3x+2=x+10→−x2x+2=10→−22x=83x + 2 = x + 10 \xrightarrow{-x} 2x + 2 = 10 \xrightarrow{-2} 2x = 8

Subtract the smaller variable term from both sides first, then solve as usual.

Checking Your Solution

Always verify your answer by substituting it back into the original equation.

x=5:2(5)−3=10−3=7✓x = 5: \quad 2(5) - 3 = 10 - 3 = 7 \checkmark

If both sides are equal after substitution, the solution is correct.