Algebra Basics: Solving 2-Step Equations - Math Antics

Summary

In this engaging Math Antics video, Rob introduces the concept of solving two-step algebraic equations that involve a combination of addition or subtraction and multiplication or division operations. He emphasizes the importance of using the Order of Operations rules in reverse to effectively undo the operations and isolate the unknown variable. Through easy-to-follow examples, Rob demonstrates the methodical approach to solving equations with and without grouping elements, highlighting the differences in outcomes based on the order of operations. The video is designed to build foundational skills for tackling more complex algebraic problems, encouraging viewers to practice and enhance their understanding.

Takeaways

• 😀 Two-step equations require undoing two operations to isolate the variable
• 😊 Use the order of operations in reverse to determine which operation to undo first
• 🤔 Parentheses create a 'group' that gets undone last when solving equations
• 🧐 The fraction line also implies grouping above and below it
• 🥸 Pay attention to how operations are grouped when solving equations
• 👓 Take one step at a time when solving multi-step equations
• 🤯 There are many possible combinations of operations in two-step equations
• 🤠 Practicing different two-step equation examples is important
• 🎓 The concepts for solving two-step equations help with more complex equations
• 🙂 Following the reverse order of operations makes solving easier

Q & A

What are the two types of math operations discussed in the video for solving equations?

The two types of math operations discussed are addition or subtraction, and multiplication or division.

Why are two-step equations considered trickier to solve than single-step equations?

Two-step equations are considered trickier because they involve more possible combinations of operations, and there's a need to decide the order in which to undo these operations.

What is the key strategy for solving multi-step equations as mentioned in the video?

The key strategy is to use the Order of Operations rules in reverse to know what order to undo operations in.

Can you give an example of a simple two-step equation discussed in the video?

A simple two-step equation discussed is: 2x + 2 = 8.

What are the inverse operations used to solve the equation 2x + 2 = 8?

The inverse operations used are subtraction and division, to undo the addition and multiplication respectively.

How does the video suggest solving an equation with both division and subtraction, such as x/2 - 1 = 4?

It suggests applying the Order of Operations in reverse: first undo the subtraction by adding 1 to both sides, then undo the division by multiplying both sides by 2.

Why is it important to consider the grouping of operations differently when solving equations?

Grouping operations differently can result in different answers, and understanding how to correctly undo operations within groups is crucial for solving the equations accurately.

What does the video say about operations inside of groups or parentheses?

Operations inside of groups or parentheses should be done first according to the Order of Operations rules, and therefore, when solving equations, operations within groups should be undone last.

What example does the video give to illustrate the concept of 'implied' groups in algebra?

The video uses the example of a fraction line automatically grouping things above or below it to illustrate the concept of 'implied' groups.

Why is practicing with different two-step equations important, according to the video?

Practicing with different two-step equations is important because it helps build understanding and familiarity with the various combinations and groupings, making it easier to solve these types of equations.

Outlines

00:00

📚 Introduction to Two-Step Equations in Algebra

Rob introduces the concept of solving two-step algebraic equations, building on the knowledge of solving simple equations from previous videos. He explains that these equations involve a combination of two operations: addition/subtraction and multiplication/division. The challenge lies in the variety of operation combinations and determining the correct order to undo these operations. Rob suggests using the Order of Operations rules in reverse as a strategy for solving two-step equations. He demonstrates this approach with an example equation, 2x + 2 = 8, explaining how to undo the addition and multiplication operations to isolate the variable 'x'. The method's effectiveness is further illustrated by solving another equation, x/2 - 1 = 4, applying the reverse Order of Operations to undo subtraction and division.

05:04

🔍 Understanding Groups in Two-Step Equation Solving

In this segment, Rob delves into the complexities of solving two-step equations involving 'groups', as indicated by parentheses or fraction lines in algebra. He emphasizes that operations within groups should be undone last, according to the reverse Order of Operations rules. This is illustrated through equations where grouping changes the approach to solving. For example, in 2(x + 2) = 8, the group (x + 2) alters the solution process compared to the earlier example. Rob explains that operations above and below a fraction line in algebra also imply grouping. He demonstrates this with a new problem, (x - 1)/2 = 4, showing how to solve it by first eliminating the division and then addressing the operation within the group. The segment underscores the importance of understanding groupings in equations to accurately apply the reverse Order of Operations.

10:04

🌟 Final Thoughts and Encouragement on Solving Two-Step Equations

Rob concludes the video by emphasizing the importance of recognizing and understanding 'implied' groups in algebra, particularly in the context of fraction lines. He reiterates that while solving two-step equations is more complex than single-step equations due to various combinations and groupings, following the reverse Order of Operations can simplify the process. Rob encourages viewers to practice solving a diverse range of two-step equations to enhance their problem-solving skills. The segment ends with Rob thanking viewers for watching Math Antics and directing them to the website for further learning, signifying the end of the video lesson.

Keywords

💡Two-step equations

Two-step equations are mathematical problems that require two operations to solve for the unknown variable. In the context of the video, these equations involve a combination of addition or subtraction and multiplication or division. The video illustrates this concept with examples, such as '2x + 2 = 8', where solving for 'x' requires both subtracting 2 from both sides and then dividing by 2, demonstrating the process of undoing operations in reverse order to isolate the variable.

💡Order of Operations

The Order of Operations is a set of rules that dictates the sequence in which mathematical operations should be performed to correctly simplify expressions. The video emphasizes the importance of these rules in both doing and undoing operations, suggesting that when solving equations, the Order of Operations should be applied in reverse. This strategy helps in determining the sequence for undoing multiple operations on an unknown variable.

💡Inverse operations

Inverse operations are pairs of mathematical operations that undo each other, such as addition and subtraction or multiplication and division. The video explains that to solve two-step equations, one must use inverse operations in a specific order. For example, to solve '2x + 2 = 8', subtraction (inverse of addition) and division (inverse of multiplication) are used to isolate 'x'.

💡Simplifying expressions

Simplifying expressions involves reducing a mathematical expression to its simplest form. The video distinguishes between simplifying expressions and solving equations, pointing out that while the Order of Operations rules apply to both, solving equations focuses on undoing operations to isolate the unknown variable. This is crucial for understanding how to approach and solve two-step equations.

💡Groups

Groups in mathematics refer to parts of an equation enclosed in parentheses or implied by the structure of the equation, indicating that operations within these groups should be performed first. The video highlights the significance of groups when solving equations, especially when they alter the approach to undoing operations. For example, a group may change the order in which operations are undone, as seen when comparing equations with and without parentheses around certain terms.

💡Fraction line

The fraction line in algebra serves not only to divide numbers but also to imply grouping of terms above and below it. This concept is crucial in the video when discussing equations involving fractions, where operations on groups defined by the fraction line are undone last. This is illustrated with the equation 'x/2 - 1 = 4', where the fraction line implies a division that must be addressed in the solution process.

💡Arithmetic operations

Arithmetic operations include addition, subtraction, multiplication, and division. The video begins by discussing equations with one arithmetic operation and progresses to those with two, emphasizing the complexity added by having multiple operations. Solving equations involves undoing these operations, and the video showcases how different combinations of these operations require specific strategies to isolate the variable.

💡Isolating the variable

Isolating the variable refers to the process of manipulating an equation so that the unknown variable stands alone on one side of the equation. This is the goal of solving equations, as demonstrated throughout the video with various examples. By using inverse operations and following the Order of Operations in reverse, the video shows how to isolate 'x' in two-step equations.

💡Mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a quantity. The video makes a clear distinction between simplifying expressions, where the Order of Operations is applied to reduce expressions to simpler forms, and solving equations, where the focus is on undoing operations to find the value of unknown variables.

💡Practice

Practice is emphasized in the video as essential for mastering the solving of two-step equations. Given the variety of equation types and the importance of understanding how to apply the Order of Operations in reverse, the video encourages viewers to solve many different problems to become proficient. This underlines the idea that solving equations is a skill developed through experience and repetition.

Highlights

Introduction to solving two-step algebra equations.

Explanation of the complexity added by having two operations in an equation.

Introduction to the concept of undoing operations to solve equations.

Discussion on the challenges of solving two-step equations due to the variety of operation combinations.

Highlighting the importance of understanding the order to undo operations.

Introduction of the idea to reverse the Order of Operations for solving equations.

Solving a simple two-step equation: 2x + 2 = 8.

Illustration of using inverse operations for solving equations.

Emphasizing the significance of the order in which operations are undone.

Solving another two-step equation with division and subtraction: x/2 - 1 = 4.

Introduction to the concept of groups in mathematical equations.

Explanation of solving equations with operations inside groups.

Solving an equation with parentheses to demonstrate the impact of groups: 2(x + 2) = 8.

Addressing the complexity added by implied groups in algebraic expressions.

Solving an equation with an implied group on the fraction line: (x - 1)/2 = 4.

Conclusion emphasizing the importance of practice and the application of the reverse Order of Operations.

Transcripts

Hi, I’m Rob. Welcome to Math Antics.

In the last two Algebra videos, we learned how to solve simple equations

that had only one arithmetic operation in them.

But often, equations have many different operations

which makes solving them a little more complicated.

In this video, we’re going to learn how to solve equations

that have just two math operations in them.

…one addition or subtraction operation, and one multiplication or division operation.

And the concepts you learn in this video will help you solve even more complicated equations in the future.

Now as you might expect, equations that have two arithmetic operations in them

are going to require two different steps to solve them.

In other words, to get the unknown all by itself, you’ll need to ‘undo’ two operations.

But that doesn’t sound too hard, right?

I mean… we learned how to do undo any arithmetic operation in the last two videos.

And that’s true. But there are a couple of reason that make two-step equations a little trickier to solve.

The first is that there are a lot more possible combinations of those two operations.

And the second is that, when there’s more than one operation, you have to decide what order to undo those operations in.

Uh…hello! If you need to know what order to do operations in,

just follow the Order of Operations rules!

You DID watch that video, didn’t you?

I sure did!

But the Order of Operations rules tell us what order to DO operations… not what order to UNDO them!

Uh…Well…then…

could we REVERSE the order since we’re UN-doing operations?

Now that’s a good idea.

Well of course it is!

When solving multi-step equations, that’s basically what we’re going to do.

Using the Order of Operations rules in reverse can help us know what order to undo operations in,

but it can be a little tricky actually putting it into practice.

So… to see how it works, let’s start by solving a very simple two-step equation: 2x + 2 = 8.

In this equation, the unknown value ‘x’ is involved in two different operations…

addition and multiplication (which is implied between the first 2 and the ‘x’)

And to undo those two operations, we need to use their inverse operations

…subtraction and division. But the question is, which one should we do first?

Like many things in life, the order we decide to do things in can make a big difference.

Ah, come on!

There’s gotta be an easier way!

[voice from off screen] “First socks, then shoes.”

Fortunately, in math, we have a special set of rules that tell us what order to do operations in.

Those rules tell us to do operations inside parentheses (or other groups) first.

And then we do exponent,

and then multiplication and division,

and last of all, we do addition and subtraction.

Those are the rules you need to follow when simplifying mathematical expressions or equations.

But solving an equation is different because we are trying to UNDO any operations

that the unknown value is involved with so that the unknown value will be all by itself.

So when solving equations, the best strategy is to apply those Order of Operations rules in reverse.

Using the reverse Order of Operations is not the only way to solve a multi-step equation,

but it’s usually the easiest way.

Just like it’s much easier to take your shoes and socks off in the reverse order that you put them on!

Ahhhhh! Are you sure it’s socks before shoes?

Since the Order of Operations rules tell us to DO multiplication before we DO addition.

We should UNDO addition before we UNDO multiplication.

So first, we undo the addition by subtracting 2 from both sides of the equation.

On the first side, the ‘plus 2’ and the ‘minus 2’ cancel each other out, leaving just ‘2x’ on that side.

And on the other side we have 8 minus 2 which is 6.

Next, we can undo the multiplication by dividing both sides of the equation by 2.

On the first side, the ‘2’s cancel, leaving ‘x’ all by itself.

And on the other side, we have 6 divided by 2 which is just 3.

There… we’ve solved the equation using the Order of Operations rules in reverse,

and now we know that x = 3.

That wasn’t so bad, was it?

Let’s try solving another simple two-step equation that has division and subtraction in it: x/2 - 1 = 4.

Again, we’re going to apply the Order of Operations rules in reverse

to undo the subtraction and the division operations.

Since we would normally DO the subtraction last, we’re going to UNDO it first.

To undo the subtraction, we add '1' to both sides of the equation.

On the first side, the ‘minus 1’ and the ‘plus 1’ cancel out, leaving just ‘x’ over 2 on that side.

And on the other side, we have 4 plus 1 which is 5.

And then, to undo the “divided by 2”, we need to multiply both sides by 2.

On the first side, the ‘2’s cancel, leaving ‘x’ all by itself.

And on the other side, we have 2 times 5 which is 10.

So our answer is x = 10.

Those examples are pretty easy, right?

But solving two-step equations gets a bit trickier

thanks to a little something in math called “groups”.

Do you remember how parentheses are used to group things in math?

And our Order of Operations rules say we are supposed to do any operations that are inside parentheses first.

In other words, we need to do operations that are inside of groups first.

Well guess what?

That means that when we’re solving equations and UN-doing operations,

we need to wait to do groups LAST of all.

To see what I mean, let’s solve this equation, which looks very similar to the first one we solved.

The only difference is that a set of parentheses has been used to group this x + 2 together.

And even though that might not seem like much of a change, it makes a big difference for our answer.

That’s because, in the original equation, this first 2 is only being multiplied by the ‘x’,

but in the new equation, it’s being multiplied by the entire quantity (or group) x + 2.

And that’s going to change how we solve it.

We’re still going to follow our Order of Operations rules in reverse,

but now that the x + 2 is inside parentheses (which means that it’s part of a group),

we’re going to undo THAT operation last.

Since we are supposed to DO operations in groups first, that means we’re going to UNDO operations in groups last.

So in this problem, we should start by undoing the multiplication that's implied between the 2 and the group (x + 2)

To do that, we divide both sides of the equation by 2.

On the first side, the 2 on the top and the 2 on the bottom cancel, leaving the group (x + 2) on that side.

And on the other side, we have 8 divided by 2 which is 4.

That looks simpler already! And we can make it even simpler than that,

because now that there’s nothing else on that side of the equal sign with the group (x + 2)

we really don’t even need the parentheses any more.

Next, we just need to subtract 2 from both sides.

On the first side, the ‘plus 2’ and the ‘minus 2’ cancel out, leaving ‘x’ all by itself,

and on the other side we have 4 minus 2, which is 2.

So for this equation, x = 2.

And now you can see how grouping operations differently in our equation results in different answers.

Let’s try one more important example.

Do you remember the second equation we solved? x/2 - 1 = 4

In this equation, the 1 is being subtracted from the entire ‘x over 2’ term.

But take a look at this slightly different equation.

This looks a lot like the original equation,

but now that the '1' is up on top of the fraction line, it’s only being subtracted from the ‘x’ and NOT the 2.

The ‘x - 1’ on top forms a group.

Hold on! How can the ‘x - 1’ be a group?

I don’t see any parentheses or brackets around it.

Ah, that’s a good question!

In Algebra, the fraction line is used as a way to automatically group things that are above it or things that are below it.

For example, in this fancy algebraic expression,

everything that’s on top of the fraction line forms a group

and everything on the bottom of the line forms another group.

Of course, we could put parentheses there if we wanted to make it really clear, but it’s not required.

Grouping above and below a fraction line is just ‘implied’ in Algebra.

Getting back to our new problem,

now that we know that the ‘x - 1’ on the top of the fraction line is an implied group,

as we learned in the last example, we’re going to wait and undo the operation inside that group last.

So the first step is to undo the ‘divided by 2’ by multiplying both sides of the equation by 2.

On the first side, the 2 on top and the 2 on the bottom will cancel out, leaving just our implied group ‘x - 1’ on that side.

And on the other side, we have 4 times 2 which is 8.

Next, we can undo the operation inside the group by adding '1' to both sides.

On the first side, the ‘minus 1’ and the ‘plus 1’ cancel, leaving ‘x’ all by itself.

And on the other side, we have 8 plus 1 which is 9. So in this equation, x = 9.

Alright… As you can see, solving two-step equations is definitely more complicated than single step equations

because there are so many different combinations and different ways to group things.

But if you just take things one step at a time and remember to UNDO operations using the REVERSE Order of Operations rules,

it will be much easier.

Just pay close attention to how things are grouped in an equation

and be on the lookout for those ‘implied’ groups on the top and bottom of a fraction line.

And, because there are so many variations of these two-step equations,

it’s really important to practice by trying to solve lots of different problems.

As always, thanks for watching Math Antics and I’ll see ya next time.

Learn more at www.mathantics.com