Question

In this unit, you solved two-step equations. A two-step equation is an equation of the form px + q = r. How could you transform this particular equation to create a three-step equation? List several ways you could transform it using common math operations. Next, try solving one of the three-step equations you created for x. (Use a guess and check strategy, if necessary.) What additional step was required to find the solution? In your opinion, is solving a three-step equation more difficult than solving a two-step equation? Why or why not? Do you think there is a limit to the number of steps it takes to solve an equation? Explain.

a) Two-step equations can be transformed into three-step equations by adding an extra variable.
b) Two-step equations can become three-step equations by introducing fractions.
c) Solving a three-step equation requires an additional step, such as multiplying or dividing by an extra number.
d) Solving three-step equations can be more challenging due to the added complexity, but there's no strict limit to the number of steps.

Answer 1

To create a three-step equation from a two-step equation, an extra operation like squaring, multiplication or division is added. Solving these equations usually entails an additional step, such as distributing a multiplied term or combining like terms, making them slightly more complex depending on the individual's proficiency in algebra.

Step-by-step explanation:

To transform a two-step equation of the form px + q = r into a three-step equation, we can employ various common mathematical operations.

• Introduce another operation such as squaring the variable term to make the equation look like px2 + q = r.
• Add a multiplication or division involving the variable, for example, px + s(x - t) = r.
• Include fractions by adding a term with a denominator, which could produce an equation such as px + ​​​​q/s = r.

Let's solve the following transformed equation step by step: 2x + 3(x - 1) = 11.

1. Distribute the 3 into the parentheses: 2x + 3x - 3 = 11.
2. Combine like terms: 5x - 3 = 11.
3. Add 3 to both sides: 5x = 14.
4. Divide both sides by 5: x = 14/5 or x = 2.8.

The additional step was combining like terms and distributing the 3, which isn't necessary in two-step equations. In terms of difficulty, three-step equations can be more challenging due to this added complexity, but it depends on the student's familiarity with algebraic manipulation.

There isn't a strict limit to the number of steps it takes to solve an equation, as complex problems may require multiple stages of solving, which may involve introducing more variables or using multiple equations. The complexity of the problem dictates the number of steps required. However, as with any problem-solving task, organizing the knowns and unknowns and selecting the right approach is crucial to efficiently find a solution.