Geo 3 4c Two Column Algebraic Proofs VIDEO

Summary

This video covers a geometry lesson on two-column algebraic proofs. The instructor guides students through three examples, demonstrating how to justify each step while solving algebraic equations. Using properties like the distributive property, combining like terms, and addition, subtraction, and division properties of equality, the goal is to prove specific values for variables. Each step of the solution process is carefully explained, helping students understand how to construct a logical argument with accepted mathematical reasoning. The lesson aims to enhance students' ability to solve algebraic equations using structured, two-column proofs.

Takeaways

• 📘 The lesson covers two-column algebraic proofs, focusing on justifying each step when solving equations.
• 📝 The first example proves that x equals 12 by starting with given information and using properties like distributive, addition, and subtraction.
• 🔢 The distributive property is applied to expand 3(4x + 5) + 3 - 7x = 90 - x in example 1, leading to a simpler equation.
• ➕ Combining like terms simplifies 12x - 7x and 15 + 3 to 5x + 18 = 90 - x.
• 🔄 The addition property of equality eliminates variables from one side by adding x on both sides.
• ➗ Division property of equality is the final step to solve x = 12 after using subtraction and division.
• 🧮 The second example involves solving 2n + 3n - 11 = 8(n - 1), using distributive and combining like terms.
• ➖ Subtracting 5n from both sides and simplifying the equation proves that n equals -1.
• 🔁 The symmetric property is used to flip the equation n = -1 to the desired form.
• ✅ The third example solves for y = -2 using the same process, with the final result proving the equation using distribution, combination, subtraction, and division.

Q & A

What is the main topic of the video?

The main topic of the video is 'Two Column Algebraic Proofs', focusing on teaching students how to justify each step while solving algebraic equations using this strategy.

What is the learning objective for the lesson?

The learning objective is for students to be able to justify each step while solving algebraic equations using the two-column proof strategy.

How many examples are discussed in the video?

There are three examples discussed in the video to illustrate the two-column algebraic proofs.

What is the purpose of the first example in the video?

The purpose of the first example is to demonstrate how to prove that x equals 12 by justifying each step with accepted reasons.

What property is used to start solving the algebraic equation in the first example?

The distributive property is used to start solving the algebraic equation in the first example.

What is the significance of combining like terms in the first example?

Combining like terms simplifies the equation by reducing the number of similar terms, making it easier to solve.

What property is used to eliminate the variable x from one side of the equation in the first example?

The addition property of equality is used to eliminate the variable x from one side of the equation by adding x to both sides.

In the second example, what property is applied to the expression '8 * n - 1'?

In the second example, the distributive property is applied to the expression '8 * n - 1' to expand it into '8n - 8'.

How does the video demonstrate the process of solving for n in the second example?

The video demonstrates solving for n by using the distributive property, combining like terms, applying the subtraction and addition properties of equality, and finally using the division property of equality.

What is the final step in proving y equals -2 in the third example?

The final step in proving y equals -2 in the third example is dividing both sides of the equation by 10, resulting in y = -2.

How does the video emphasize the importance of starting and ending with given information?

The video emphasizes the importance of starting with given information and ending with what is to be proved by showing that each step is justified and that the process begins and concludes with the given information.

Outlines

00:00

📐 Introduction to Two-Column Algebraic Proofs

This video script introduces a geometry lesson focused on two-column algebraic proofs, specifically lesson 3.4 C. The objective is for students to learn how to justify each step while solving algebraic equations using this proof strategy. The lesson is found on page 12 of the chapter 3 packet. The script outlines three examples to demonstrate the process, starting with rewriting the given equation and then using the distributive property to simplify. The goal is to prove that x equals 12 by justifying each step with accepted reasons. The first example involves simplifying the equation 3 * 4x + 5 + 3 - 7x = 90 - x, and the process includes combining like terms and applying properties of equality to isolate x.

05:01

🔍 Detailed Steps in Solving Algebraic Equations

The second paragraph delves deeper into the process of solving algebraic equations with a focus on proving that n equals -1. It starts with the given equation 2n + 3n - 11 = 8n - 1 and applies the distributive property to simplify. The script then instructs viewers to combine like terms, resulting in 5n - 11 = 8n - 8. The next steps involve using the subtraction and addition properties of equality to isolate n, eventually leading to the conclusion that n = -1. The paragraph emphasizes the importance of starting with given information and ending with the proof, showcasing the logical flow of an algebraic proof.

10:04

📘 Final Example and Conclusion

The final paragraph presents the last example, aiming to prove that y equals -2. It begins with the given equation 5y + 4 - 19 = 5 * (3y + 1) and proceeds to apply the distributive property. The script then instructs viewers to combine like terms and use the subtraction property of equality to simplify the equation. The symmetric property is used to flip the equation, and the final steps involve subtracting and dividing to isolate y, ultimately proving that y = -2. The video concludes by thanking viewers for watching and summarizing the key points covered in the lesson.

Keywords

💡Two-Column Algebraic Proof

A two-column algebraic proof is a structured method used in geometry and algebra to demonstrate the validity of an equation or statement. It involves breaking down each step of the solution process into two columns: 'Statements' and 'Reasons.' Each statement explains what is being done, and each reason justifies why it can be done. This approach ensures clarity and logical flow, as seen in the video when solving equations and proving equalities.

💡Statement

In a two-column proof, a 'statement' refers to the specific mathematical operations or simplifications made in each step. For example, when solving an equation, a statement might involve applying the distributive property or combining like terms. Statements must be presented in a logical order to lead to the final conclusion, such as proving that 'x = 12.'

💡Reason

A 'reason' in a two-column proof justifies the corresponding statement. Reasons are typically based on mathematical properties, such as the distributive property, addition property of equality, or combining like terms. Each reason explains why a step is valid, supporting the logical argument. In the video, reasons like 'subtraction property of equality' are used to justify subtracting the same value from both sides of an equation.

💡Distributive Property

The distributive property is a fundamental algebraic principle that allows multiplication to be distributed over addition or subtraction. It is represented as a(b + c) = ab + ac. In the video, this property is applied to expand expressions like '3(4x + 5)' into '12x + 15,' making it easier to solve complex equations step-by-step.

💡Combining Like Terms

Combining like terms involves simplifying an expression by adding or subtracting terms that have the same variables raised to the same power. For instance, '12x - 7x' is simplified to '5x.' This technique is crucial for reducing the complexity of equations and is used multiple times in the video to streamline the solving process.

💡Addition Property of Equality

The addition property of equality states that if the same amount is added to both sides of an equation, the equality remains true. In the video, this property is used when 'x' is added to both sides of an equation to maintain balance and eliminate variables from one side, making the equation easier to solve.

💡Subtraction Property of Equality

The subtraction property of equality is similar to the addition property but involves subtracting the same quantity from both sides of an equation. This property helps isolate variables and simplify equations, as demonstrated in the video when 5y is subtracted from both sides to transform '5y - 15 = 15y + 5' into a simpler form.

💡Division Property of Equality

The division property of equality states that dividing both sides of an equation by the same non-zero number preserves equality. In the video, this property is applied in the final steps of solving equations, such as dividing both sides by 6 to isolate 'x' and prove that 'x = 12.' It is essential for reaching the final solution of many algebraic problems.

💡Symmetric Property

The symmetric property of equality states that if 'a = b,' then 'b = a.' This property allows the order of an equation to be switched for clarity or preference. In the video, it is used to transform 'n = -1' into '-1 = n' to match the desired proof format, demonstrating flexibility in presenting equalities.

💡Proof

A proof is a logical argument that establishes the truth of a statement using a sequence of justified steps. In the context of the video, proofs are used to demonstrate that certain algebraic equations lead to specific results, such as proving 'x = 12' or 'n = -1.' Each step in a proof builds upon the previous one, ensuring that the conclusion is reached through sound reasoning and accepted mathematical principles.

Highlights

Introduction to two column algebraic proofs

Learning objective: justifying each step while solving algebraic equations

Example 1: Proving x equals 12 using given equation

Use of distributive property in algebraic proofs

Combining like terms to simplify equations

Applying addition property of equality to eliminate variable

Subtraction property of equality to simplify further

Division property of equality to solve for x

Example 2: Proving n equals -1 with given equation

Applying distributive property to parentheses

Combining like terms to simplify the equation

Subtraction property of equality to isolate variable

Addition property of equality to simplify further

Division property of equality to solve for n

Symmetric property to switch the order of equality

Example 3: Proving y equals -2 with given equation

Distributive property applied to prove y equals -2

Combining like terms to simplify the equation

Subtraction property of equality to isolate variable

Symmetric property to rearrange the equation

Subtraction property of equality to further simplify

Division property of equality to solve for y

Conclusion and thank you for watching

Transcripts

hello everybody this video is for

geometry lesson 3.4 C the topic for this

lesson is two column algebraic proofs

and you can find this on page 12 in your

chapter 3 packet the learning objective

for today is that students will be able

to justify each step while solving

algebraic equations using the two column

proof

strategy all right we have three

examples for today to talk through

and um a few blanks to fill in here so

3.4 C2 column algebraic proofs a proof

is a logical

argument in which each

statement is supported by a statement

that is accepted as

true and so the idea here is that we're

going to go from this given information

in this problem and we're going to prove

that x equal 12 at the end and we're

going to justify all of our statements

with reasons that are accepted as true

all right so remember statement one and

reason one are always going to include

the given

information and so what I would like you

to do is rewrite this given equation

here

and we're going to write this right here

rewrite that in that blank all right so

we've got 3 * 4x + 5 + 3 - 7 x = 90 - x

now if we're solving this algebraic

equation we would likely start with the

distributive

property and that's going to be our

reason for this step so we're going to

distribute

and when we distribute here we're going

to get

12x +

15+

3 - 7

x = 90 -

x all right so that's our distributed

Distributing

property after this we were going to

look at our left side here and we're

going to combine some like terms

we're going to combine like

terms all right so what can we combine

we can combine 12x with -7x that gives

us 5x we can combine 15 plus 3 that's

going to give us+ 18 and that's Phil

equal to 90 - x over here on the

right all right after this we're going

to use our normal steps um and I would

recommend that we add X on both sides

that way we eliminate X's from one of

the sides and this is going to be our

addition property of

equality addition property of equality

and what we get over here now is 5x + x

is 6 x +

18 = 90 and now this equation starting

to look like a more simple equation to

solve we have two steps to go we're

going to take away subtract

18 and that's our subtraction property

of

equality and that's going to give us

6X =

72 and our last step would be divide by

six on both sides that's our division

property of equality

and that's going to give us what we're

trying to prove which is that x = 12 at

the very end and just to point out a

couple of things here so I want us to

notice that we started with our given

information and that's what we had in

statement one and reason one and then we

ended with what we're trying to prove

which is what we ended with down here in

statement six all right so that's

example one for

today example number two all right so a

little bit more filling in the blank for

this one so we're given that 2 n + 3 N -

11 = 8 n 8 * N - 1 and we want to prove

that n equal

-1 all right so I noticed that in the

given spot here we have 2 n + 3 N - 11 =

8 * n -1 that's our given

information and remember the first

reason should always be

given we're going to use a few

properties here and I just want to point

out what we're trying to do here is

prove that

nal1 and so I'm going to go ahead and

write nal1 at the end because that's

what we're trying to prove and that's

what we should end up with as our very

last

step all right now we don't have too

much to do here but we're going to start

with the distributive property it says

distributive property right here and the

only spot where we can apply that is to

these parentheses 8 * n -1 so that gives

us 8 N - 8 when we Use the distributive

property and then everything else

Remains the Same 2N + 3 N -

11 all right so that's our new equation

now how do we go from statement two to

statement three well I notice that the

everything stays the same except for the

2N plus the 3 n became a

5n and that is combining like terms

terms so I'd like you to write in combin

like terms

here so this step we combined like terms

to get 2 n + 3 n = 5n and now we're

going to use our normal steps to solve

so normally from here we would take away

5n from both

sides and that is our subtraction

property of equality because we're

taking away we're subtracting

5n subtraction property of of

equality and that will give us -11 = 3

nus 8 after this we're going to use the

addition property so that would be + 8 +

8 on both

sides and we will get -3 = 3

N8 + 8 cancel out and we get -3 = 3 n

now to go from 5 to six we would have to

divide by three on both sides to get n

uhga -1 equals n and that's our division

property of

equality because we're dividing both

sides by the same

number and then lastly and this um is

just so that we can switch the order but

we our result here is -1 = n but we

wanted to prove that n equal -1 and so

when we take the same statement and we

just flip the order here that's using

our symmetric property and so we end

with Nal um

nal1 all right final example for this

page all right so we are given that 5 y

+ 4 - 19 = 5 * in parentheses 3 y + 1

and we're going to prove that y = -2 I

want you to notice that our given

information here is already in state

statement and reason one and I want to

point out that our what we're trying to

prove y = -2 is already down here in

statement 7 all right so let's go step

by step we're going to start with the

distributive property it says

distributive property here and so we're

going to distribute our where we see

parentheses and we'll get 15 y +

5 and then on this side it Remains the

Same 5 y + 4 - 19

after this step we're going to combine

like terms and I see some like terms

over here that we can combine we have

pos4 - 19 that gives us

15 and then let's rewrite everything

else so 5 y - 15 = 15 y +

5 and then from step three to step 4 I

notied that the only thing that's gone

is the 5 y on this side so we're going

to subtract 5 y from both

sides and we get -15 = 10 y + 5 and

because we subtracted on both sides

that's going to be our

subtraction property of

equality now we're going to use the

symmetric property in step five and

remember symmetric property just means

we flip both

sides so the left becomes the right and

the right becom comes left so we're just

going to flip these and we get 10 y + 5

= -15 and you could do this at a later

time but this is a convenient time

because we know that Y is equal y equals

2 at the end and the Y is on the right

side here it's on the left side here so

at some point we'll have to flip it all

right now our last two steps we're going

to subtract five on both sides that's

our subtraction

property and we get 10 y =

-20 and our last step division property

divide by 10 divide by 10 and we get y =

-2 which is what we were looking to

prove originally thank you for watching

this video that's all for these three

examples thank you