which equation is equivalent to

Daniel Parker

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28-02-2025

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Which Equation is Equivalent? Mastering Equivalent Equations

Finding equivalent equations is a fundamental concept in algebra. Understanding how to manipulate equations while maintaining their equality is crucial for solving problems and understanding mathematical relationships. This article will explore various methods for determining equivalent equations and provide examples to solidify your understanding. The core idea is that equivalent equations have the same solution set; that is, the same value(s) of the variable(s) make both equations true.

What Makes Equations Equivalent?

Two equations are considered equivalent if they have the same solution(s). This means any value of the variable that satisfies one equation will also satisfy the other. We can create equivalent equations using several algebraic operations:

• Adding or Subtracting the Same Value: Adding or subtracting the same number or expression to both sides of an equation maintains the equality.
• Multiplying or Dividing by the Same Non-Zero Value: Multiplying or dividing both sides by the same non-zero number or expression also preserves the equality. Dividing by zero is undefined, so it's crucial to avoid this.
• Simplifying Expressions: Combining like terms, distributing, or factoring can simplify an equation without changing its solution set, making it equivalent to the original.

Identifying Equivalent Equations: Examples

Let's examine some examples to illustrate how to identify equivalent equations:

Example 1:

Is the equation 2x + 4 = 10 equivalent to x + 2 = 5?

To check, we can solve both equations:

• 2x + 4 = 10: Subtract 4 from both sides: 2x = 6. Divide by 2: x = 3.
• x + 2 = 5: Subtract 2 from both sides: x = 3.

Both equations have the same solution (x = 3), so they are equivalent. Notice that the second equation is derived from the first by dividing both sides by 2.

Example 2:

Are 3x - 6 = 9 and 3x = 15 equivalent?

Again, let's solve:

• 3x - 6 = 9: Add 6 to both sides: 3x = 15. Divide by 3: x = 5.
• 3x = 15: Divide by 3: x = 5.

Both equations have the solution x = 5, confirming they are equivalent. The second equation is obtained by adding 6 to both sides of the first.

Example 3: A More Complex Scenario

Consider the equations: 2(x + 3) = 10 and 2x + 6 = 10. Are they equivalent?

Let's solve:

• 2(x + 3) = 10: Distribute the 2: 2x + 6 = 10. Subtract 6 from both sides: 2x = 4. Divide by 2: x = 2.
• 2x + 6 = 10: Subtract 6 from both sides: 2x = 4. Divide by 2: x = 2.

Both equations yield the same solution (x = 2), confirming their equivalence. The second equation is simply the result of distributing the 2 in the first equation.

Example 4: Non-Equivalent Equations

Not all equations that look similar are equivalent. Consider:

x + 2 = 5 and x² + 2x = 5x

While they might seem related, they have different solution sets. Solving them reveals this difference:

• x + 2 = 5: x = 3
• x² + 2x = 5x: Rearrange to x² - 3x = 0, factor to x(x-3) = 0. This gives solutions x = 0 and x = 3.

These equations are not equivalent because they have different solution sets.

Common Mistakes to Avoid

• Dividing by a variable: Never divide both sides of an equation by a variable unless you know it's non-zero. This can lead to losing solutions.
• Incorrect simplification: Careless algebraic manipulation can lead to non-equivalent equations. Double-check your steps.

Mastering the identification of equivalent equations is a key skill in algebra. By understanding the permissible operations and carefully applying them, you can confidently manipulate equations and solve various mathematical problems. Remember, the hallmark of equivalent equations is the shared solution set.