Solving One-Step Integer Equations

Lesson Features »

Lesson Priority: High

 
Objectives
  • Recognize the two types of one-step equations
  • Solve addition and subtraction one-step equations
  • Solve multiplication and division one-step equations
Lesson Description

In this introductory equation solving lesson, you'll learn how to solve the two different kinds of one-step linear equations.

 
Practice Problems

Practice problems and worksheet coming soon!

 

One and Done

As mentioned in the last lesson about properties of equations », one of the things we should remember about equations is that if we "do the same stuff" to both sides of an equation, the equation is still valid. This first lesson of many on solving equations will show us how to use that property to figure out an equation's solution, for cases where we only have to do one action to get the answer.
Pro Tip
In this lesson and every lesson about solving equations, our goal will be to "isolate" the variable by performing opposite actions. Every category of equations we solve will have its own lesson to showcase particular patterns, but it's helpful to realize right away that all equation solving in math has the same goal.
The general idea is that we will look at the equation given to us and see what it being done to the variable. Then we will apply the opposite arithmetic to both sides, which will "undo" what was being done to the variable, and it will be alone. When it is alone, we can see what its value is that makes the equation true.There are two types of one-step equations we'll study here: one for adding and subtracting, and another multipying and dividing.

One-Step Type 1: Add / Subtract

Equations that only require adding or subtracting will have a variable with a coefficient of $1$ and a constant that is added or subtracted to it. If a number is added to the variable, subtract that number from both sides. If a number is subtracted from the variable, add it to both sides.Let's see two examples.
 
Example 1$$x + 4 = 7$$$\blacktriangleright$ Since $4$ is being added to $x$, we need to subtract $4$ from each side of the equation.$$\begin{align} x+4 &= 7 \\ -4 & \,\, -4 \end{align}$$$$x + 4 - 4 = 7 - 4$$$$x = 3$$Having got $x$ alone on one side of the equation, we can look at the resulting open equation » and ask ourselves what value of $x$ would make the equation $x=3$ true - hopefully realizing quickly that it is indeed $3$. That's the point of isolating the variable - it's no longer a challenge to solve an open equation when the variable is isolated. The answer is staring back at us.
Warning!
In the above example and in many one-step cases, you might have looked at the original given equation and known instantly that $3$ was the solution to it. That's great! Don't interpret what is being said in this lesson to mean that these equations are unsolvable without following "proper protocol". However, when you're starting out with this stuff, your teacher's going to want to see a few pen strokes of scratch work on tests. Additionally, when you start working on things like$$3(n+4) - 2n -9 = 7 + 2(2-n)$$you will not be able to look at the equation and know the solution. And if you didn't practice the boring basics right from the start, you'll have a harder time when you need to use those steps!
 
Example 2$$x - 9 = -3$$$\blacktriangleright$ Since $9$ is being subtracted from $x$, we need to add $9$ to both sides to isolate $x$.$$x-9+9 = -3 + 9$$$$x = 6$$The result we get tells us that $x$ is $6$ in order for the original equation to be true.

One-Step Type 2: Multiply / Divide

Equations that only require multiplying or dividing will have a variable with a coefficient that is not $1$ and no constant added or subtracted to it. Depending on the coefficient, we will multiply or divide to "undo" it.
 
Example 3$$ 5n = 45$$$\blacktriangleright$ In words, this equation says "$5$ times a number $n$ is $45$". The variable in this equation is being multiplied by $5$. To undo this action, we should divide both sides of the equation by $5$.$$\frac{5n}{5} = \frac{45}{5}$$$$\longrightarrow n = 9$$The solution to our equation is $n=9$. This is the value of $n$ which, when substituted into the original equation, makes the equation true.
Pro Tip
When you multiply or divide both sides of an equation, it is important to realize that we are doing that action to the entirety of each side of the equation. This distinction is less consequential now than it will be in the future, but knowing and understanding that from the get-go will pay off later on.It's also worth pointing out that we'll often use fraction symbols to represent division, instead of the old-school $\div$ sign. The fraction notation is much more convenient for manipulating equations as we get further down the Algebra rabbit hole.
 
Example 4$$\frac{x}{4} = 7$$$\blacktriangleright$ This equation has the variable being divided by $4$ on the left side. If we would like to "undo" that operation, we need to multiply by $4$ on both sides.$$4 \cdot \left[ \frac{x}{4} \, \right] = 4 \cdot [7]$$$$x = 28$$The solution to the equation is $x=28$.

Put It To The Test

Let's run through one more of each of the four types of one-step integer equations. If you want to further perfect your solving skills, make sure to grab the accompanying worksheet for this lesson!
You Should Know
These equations are the most fundamental to solve because they only require one step. If your teacher is moving quickly or trying to multi-task, you may be learning how to solve two-step equations » at the same time, which we'll look at in the very next lesson.
 
Example 5$$x-6 = 4$$
Show solution
$\blacktriangleright$ Add 6 to both sides.$$x-6+6 = 4+6$$$$x = 10$$
 
Example 6$$3x=57$$
Show solution
$\blacktriangleright$ Divide both sides by $3$.$$\frac{3x}{3} = \frac{57}{3}$$$$x = 19$$
 
Example 7$$x+8 = 8$$
Show solution
$\blacktriangleright$ Subtract 8 from both sides.$$x+8-8 = 8 - 8$$$$x = 0$$
 
Example 8$$\frac{x}{-3} = 5$$
Show solution
$\blacktriangleright$ Multiply both sides by $-3$.$$(-3) \cdot \left[ \frac{x}{-3} \, \right] = (-3) \cdot 5$$$$x = -15$$
 
Lesson Takeaways:
  • Understand what it means to "isolate" the variable, and why that's a very helpful step in equation solving
  • Practice correctly applying one step arithmetic to both sides of an equation to solve for the variable
  • For multiplication and division, understand that the "thing" you are actually doing is multiplying or dividing the whole side
  • For division, gain comfortability with the fraction bar notation for representing division, instead of the old-school $\div$ symbol

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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