2 Step Equations That Equal 1

Solving Two-Step Equations That Equal 1: A Comprehensive Guide

Solving equations is a fundamental skill in algebra. While seemingly simple, mastering the technique is crucial for tackling more complex mathematical problems. This guide dives deep into solving two-step equations that equal 1, covering various scenarios, common pitfalls, and strategies for success. We’ll explore different equation types, offer step-by-step solutions, and provide practice problems to solidify your understanding.

Understanding Two-Step Equations

A two-step equation involves two operations performed on a variable. These operations can be addition, subtraction, multiplication, or division. The goal is to isolate the variable (usually represented by 'x' or another letter) on one side of the equation to find its value. For example, a typical two-step equation might look like this: 2x + 3 = 7. In this case, 'x' is multiplied by 2 and then 3 is added. To solve, we must reverse these operations in the correct order.

When our target solution is 1, it adds an interesting dimension. This means that after applying our two-step solution, the variable will have a value that results in a final answer of 1. Let’s explore this in detail.

Step-by-Step Approach to Solving Two-Step Equations Equal to 1

The key to solving two-step equations, regardless of the target value, lies in using inverse operations. Inverse operations are operations that “undo” each other. Addition and subtraction are inverses, as are multiplication and division.

The General Strategy:

1. Isolate the term with the variable: This usually involves adding or subtracting a constant from both sides of the equation.
2. Isolate the variable: This involves multiplying or dividing both sides of the equation by the coefficient of the variable.

Let's illustrate this with examples:

Example 1: 3x + 2 = 5

1. Subtract 2 from both sides: 3x + 2 - 2 = 5 - 2 => 3x = 3
2. Divide both sides by 3: 3x / 3 = 3 / 3 => x = 1

In this case, the solution is x = 1. This is a straightforward example where the final solution equals 1.

Example 2: 2x - 5 = -3

1. Add 5 to both sides: 2x - 5 + 5 = -3 + 5 => 2x = 2
2. Divide both sides by 2: 2x / 2 = 2 / 2 => x = 1

Again, x = 1. This shows how different starting equations can lead to the same solution.

Example 3: (x/4) + 1 = 1.25

1. Subtract 1 from both sides: (x/4) + 1 - 1 = 1.25 - 1 => x/4 = 0.25
2. Multiply both sides by 4: (x/4) * 4 = 0.25 * 4 => x = 1

Here, we used decimals, demonstrating the applicability of the method across different number types. The solution, however, remains x = 1.

Example 4: Incorporating Negative Numbers -7x + 14 = 7

1. Subtract 14 from both sides: -7x + 14 - 14 = 7 - 14 => -7x = -7
2. Divide both sides by -7: -7x / -7 = -7 / -7 => x = 1

Note how handling negative numbers requires careful attention to signs.

Common Mistakes to Avoid

Several common mistakes can hinder the process of solving two-step equations. Being aware of these can significantly improve accuracy and efficiency.

• Incorrect Order of Operations: Always perform addition or subtraction before multiplication or division when isolating the variable. Reversing this order will lead to incorrect results.

• Errors with Signs: Pay close attention to the signs of the numbers. Adding a negative number is the same as subtracting a positive number, and vice versa. Similarly, dividing by a negative number changes the sign of the result.

• Fractional Coefficients: Equations with fractional coefficients can be challenging. Remember that multiplying by the reciprocal of a fraction is equivalent to dividing by the fraction. For instance, if you have (1/3)x = 2, multiplying both sides by 3 will isolate 'x'.

• Distribution Errors: If your equation involves parentheses, remember to distribute any coefficients before proceeding with the other steps. For example, 2(x + 1) = 4 should become 2x + 2 = 4 before further solving.

Advanced Scenarios and Variations

Let's look at some more intricate examples that build upon the fundamental concepts already discussed.

Example 5: Equations with Decimals 0.5x + 0.75 = 1.25

1. Subtract 0.75 from both sides: 0.5x + 0.75 - 0.75 = 1.25 - 0.75 => 0.5x = 0.5
2. Divide both sides by 0.5: 0.5x / 0.5 = 0.5 / 0.5 => x = 1

This demonstrates handling decimal values gracefully.

Example 6: Equations with Fractions (2/3)x - (1/2) = 1/6

1. Add (1/2) to both sides: (2/3)x - (1/2) + (1/2) = (1/6) + (1/2) => (2/3)x = (2/3)
2. Multiply both sides by (3/2): (3/2) * (2/3)x = (2/3) * (3/2) => x = 1

This showcases a more complex equation with fractions, requiring a clear understanding of fraction arithmetic.

Practice Problems

To reinforce your understanding, let's tackle some practice problems:

1. 5x + 4 = 9
2. 7x - 10 = -3
3. (x/2) - 3 = -2.5
4. -4x + 8 = 4
5. 0.25x + 0.5 = 0.75
6. (3/4)x - (1/4) = 1/2
7. -2(x - 3) = 4
8. 1/5x + 2/5 = 3/5

Solutions: (Check your work!)

1. x = 1
2. x = 1
3. x = 1
4. x = 1
5. x = 1
6. x = 1
7. x = 1
8. x = 1

Conclusion

Solving two-step equations that equal 1, or any value for that matter, is a crucial algebraic skill. By understanding the inverse operations, following a systematic approach, and being mindful of common mistakes, you can master this technique. Consistent practice, working through various examples, and tackling problems with different numbers and complexities will solidify your understanding and build your confidence in tackling more challenging algebraic problems in the future. Remember, the key lies in systematic application of inverse operations and careful attention to detail.