Solving Two-Step Integer Equations

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Lesson Priority: High

 
Objectives
  • Recognize the general form of a two-step equation
  • Solve two step-equations and show the right work
Lesson Description

Two-step equations are probably the most common type of equation we work with throughout Algebra. In this lesson we build upon what we learned in the last lesson, and master solving two-step linear equations.

 
Practice Problems

Practice problems and worksheet coming soon!

 

A Common Two-Step Dance

Solving Equations is one of the tasks we spend the most effort working toward in math. In fact, many of the small skills we pick up along the way are learned with the intention that they will be used in some way to help us solve equations in new ways. Now that we've learned how to solve both kinds of one-step equations, we are ready to solve similar equations that require two steps - a very common and important process which will pave the way for new equation solving skills in the near future.A Balancing ActAs we've discussed recently (and will continue to discuss), the nature of equations requires us to look at them with the perspective of balancing. Because each side of an equation is "equal" to one another, that means that each side is the same value, even if each side's appearance is different. When we say $3x = 9$, we literally mean that $3x$ and $9$ are the same things, even though they physically look different. It's like having four 5 dollar bills or having two 10 dollar bills - the appearance is different but each is worth 20 bucks regardless.Since equations amount to having two quantities that are equivalent, our manipulation techniques need to reflect balance. This refers to the idea that whatever we do to one side, we do to the other. If $3x$ and $9$ have the same value, then $3x+1$ and $9+1$ do as well. Now let's get down to business on two-step equations.

Form of a Two-Step Equation

Generally, a two-step one variable equation looks like this:$$ax + b = c$$where $a$, $b$, and $c$ are some numbers. For example,$$2x - 5 = 9$$$$-x + 11 = 4$$$$3x - 4 = -22$$whereas something like$$4(x+6) - 2x = 5x +3$$is not a two-step equation (though it is a multi-step one variable equation, which we will practice solving in the next lesson »). The general thing that makes a one variable equation of the two-step variety is
  • one term in the equation is the variable term
  • the side of the equation that has the variable term has another number added or subtracted to it
  • the other side of the equation just has a number
If the equation is any more complex, it requires more than two steps to solve ». If the equation is any less complex, it would be one of the one-step equations » that we recently learned about, either with addition and subtraction or with multiplication and division.Why are we making a big deal about recognizing the form? In truth it's not a big deal. Once you master these one variable, non-exponent equations, including the upcoming lesson on multi-step equations », you'll not need to tell them apart - you'll be very fluent in the steps to solve any of these equations. For now, let's just try and understand why these two-step equations are indeed solvable in two-steps, and know that if you're still relatively new to equation solving, identifying and mastering the two-step equation will do you very well for continuing to understand general concepts about equation solving.
Pro Tip
Knowing various forms of these early-Algebra equations (such as one-step, two-step, multi-step, etc) is nice, but not required. You'll gain fast confidence in solving any of these equations with some solid practice. Identifying their classification is useful as we first learn and understand these situations, but it isn't going to add as much value after you become the one-variable equation master.

Solving Two-Steppers

Now we turn to the structured, repeatable process for solving these types of equations. Our example will be$$4x - 3 = 33$$Step 1 - Isolate the Variable TermTake a look at whichever side of the equation that has the variable term. We want to isolate this variable term by adding or subtracting (on both sides, of course) the standalone number that is also on that side of the equation. We often refer to this as "moving" the constant. In our example, we have$$4x - 3 = 33$$so we want to add $3$ to both sides:$$4x - 3 = 33$$$$\,\,\,\,\,\,\,\,\, + 3 \,\,\,\,\,\, +3$$$$4x - 3 + 3 = 33 + 3$$$$\Longrightarrow 4x = 36$$Step 2 - Multiply or DivideThe second and final step will be to "neutralize" the constant that is attached to your variable. In order to do this divide by the constant. If you prefer (as we sometimes will), you can think of this as multiplying by the reciprocal of the constant.$$4x = 36$$$$\longrightarrow \frac{4x}{4} = \frac{36}{4}$$$$\boxed{x = 9}$$We can say that this step was to divide both sides by the constant $4$, or we could equivalently say that we multiplied by sides by the reciprocal of $4$ (which is $1/4$). Again, depending on whether fractions are already present in a problem, we may find it more convenient to take one perspective or the other. For the time being, we will focus on integer based equations, saving fraction coefficients for a later lesson ».
You Should Know
Step 2 is really just the same process that we learned in the previous lesson » for solving multiplication and division style one-step equations. This is because, upon completing Step 1, you've essentially turned the original problem into a one-step equation of the form that we saw in the last lesson.
You will be able to follow this simple, repetitive, prescriptive approach for every equation that you would be able to classify as "two-step", the way that we defined "two-step" at the start of this lesson. If your problem is simpler, it's probably just one-step, which will require similar but easier steps. If your problem looks more complex, you should check out the lesson immediately following on Multi-Step Equations ».
Pro Tip
A common pitfall for students as they start out learning equation solving techniques is to figure out the answer mentally, write the answer, and omit your scratch work. This is especially easy to do when the equations are one-step or two-step - DON'T let this happen to you! You are setting yourself up to lose points, and even worse, starting a bad habit that's hard to kick. Solving equations quickly becomes much more challenging in Algebra, so don't think you'll be able to mental math your way to success for your whole career. Write every step out, and write both sides of the equation every time you do it.

Put It To The Test

The best route here is simply practice. This process is rigid and once you have it down, you'll not only be ready for similar problems forever more, but also in good shape to tackle multi-step equations next. Work through the problems below and check your work as you go - make sure you're expertly for dealing with negative terms and coefficients!
 
Example 1$$2x-8 = 4$$
Show solution
$$\blacktriangleright \,\, 2x - 8 + 8 = 4 + 8$$$$\longrightarrow 2x = 12$$$$\longrightarrow \frac{2x}{2} = \frac{12}{2}$$$$\boxed{x=6}$$
 
Example 2$$5x + 3 = -17$$
Show solution
$$\blacktriangleright \,\, 5x + 3 - 3 = -17 - 3$$$$\longrightarrow 5x = -20$$$$\longrightarrow \frac{5x}{5} = \frac{-20}{5}$$$$\boxed{x=-4}$$
 
Example 3$$9 = -7 + 4x$$
Show solution
$\blacktriangleright$ Don't let the reverse order of terms make you think that this situation is any different - it is two-step because it meets the criteria (a single variable term accompanied by a constant, and a lone constant on the other side). The steps are the same: isolate the variable term, then neutralize the coefficient.$$9 +7 = -7 + 4x +7$$$$\longrightarrow 16 = 4x$$$$\longrightarrow \frac{16}{4} = \frac{4x}{4}$$$$\boxed{x=4}$$
 
Example 4$$-3x + 5 = 50$$
Show solution
$$\blacktriangleright \,\, -3x + 5 - 5 = 50 - 5$$$$\longrightarrow -3x = 45$$$$\longrightarrow \frac{-3x}{-3} = \frac{45}{-3}$$$$\boxed{x=-15}$$
Remember!
Don't be intimidated by negative coefficients or constants. Just follow the approach, and after a handful of practice problems, you'll be well-prepared to get these questions consistently correct regardless of whether your teacher tries to throw you off with negatives.
 
Lesson Takeaways
  • Be able to recognize the form of a two-step equation
  • Know the two steps to solving such equations, and understand why the order of the steps is important
  • Show all scratch work for every problem you write, even if it's a simple problem, and even if you know the answer mentally

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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