One-Step and Two-Step Equation Worksheets: Inverse Operations

Algebra and Function Practice

Solving an equation means finding the value of the variable that makes the statement true. For linear equations that need one or two inverse operations, the goal is to undo operations in reverse order while keeping the equation balanced. This lesson provides a consistent workspace: left column for the equation and each inverse operation, right column for the resulting simplified line. The convention is to show the operation applied to both sides on the same line, then simplify. Numerical examples are hypothetical and chosen for easy arithmetic; always measure your own equipment if printing or handwriting constraints matter.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

One-Step Equations: Undo a Single Operation

A one-step equation contains one operation acting on the variable, such as addition, subtraction, multiplication, or division. To isolate the variable, apply the inverse operation to both sides. For x + 7 = 15, subtract 7 from both sides to get x = 8. For 4x = 20, divide both sides by 4 to get x = 5. For x/3 = 6, multiply both sides by 3 to get x = 18.

Keep the workspace tidy: write the original equation, then on the next line write the inverse operation applied to both sides, then the simplified result. Check by substituting the value back into the original equation.

1. Identify the operation acting on the variable.
2. Choose the inverse operation: addition undoes subtraction, multiplication undoes division, and vice versa.
3. Apply the inverse operation to both sides of the equation.
4. Simplify both sides to isolate the variable.
5. Substitute the value back into the original equation to verify.

Two-Step Equations: Reverse the Order

A two-step equation has two operations acting on the variable, such as 3x + 4 = 19. Undo operations in reverse order of standard arithmetic: first undo addition or subtraction, then undo multiplication or division. For 3x + 4 = 19, subtract 4 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5.

For equations like x/2 - 5 = 1, first add 5 to both sides to get x/2 = 6, then multiply both sides by 2 to get x = 12. Always keep the equation balanced by performing the same operation on both sides.

1. Identify the two operations acting on the variable.
2. Undo addition or subtraction first by applying the inverse operation to both sides.
3. Simplify to obtain a one-step equation.
4. Undo multiplication or division by applying the inverse operation to both sides.
5. Simplify to isolate the variable and check by substitution.

Workspace Convention and Handwriting Notes

Use a two-column workspace. Left column: write the equation, then each inverse operation on its own line, showing the operation applied to both sides. Right column: write the simplified result after each step. For example, left column: 2x + 3 = 11; subtract 3 from both sides; 2x = 8; divide both sides by 2. Right column: x = 4.

Handwriting varies: x-height is measured from baseline to midline, and letter styles differ. Accept any legible style that clearly distinguishes the variable from operators and numbers. The rubric below allows valid alternatives rather than a single universal answer.

1. Set up two columns: left for operations, right for simplified lines.
2. Write the original equation at the top of the left column.
3. On each new line, state the inverse operation applied to both sides.
4. Record the simplified equation in the right column.
5. Circle or box the final isolated variable.

Worked example

Solve 5x - 7 = 18 using inverse operations and show each step.

Four practice exercises

Common mistakes

• Applying the inverse operation to only one side of the equation. Always apply the same operation to both sides to keep the equation balanced.
• Undoing multiplication before addition in a two-step equation. Reverse the order: undo addition or subtraction first, then multiplication or division.
• Confusing the inverse of division with subtraction. The inverse of division is multiplication; the inverse of subtraction is addition.
• Skipping the substitution check. Substitute the solution into the original equation to confirm both sides are equal.

Self-check rubric

• Correctly identifies the operation(s) acting on the variable.
• Applies the inverse operation to both sides at each step.
• Undoes addition or subtraction before multiplication or division in two-step equations.
• Shows a clear two-column workspace with operations and simplified lines.
• Substitutes the solution back into the original equation to verify.
• Handwriting is legible; x-height from baseline to midline is consistent enough to distinguish the variable, and valid alternative letter styles are accepted.

Questions and answers