Geo 3 4c Two Column Algebraic Proofs VIDEO
Summary
TLDRThis 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
- ๐ The lesson covers two-column algebraic proofs, focusing on justifying each step when solving equations.
- ๐ The first example proves that x equals 12 by starting with given information and using properties like distributive, addition, and subtraction.
- ๐ข The distributive property is applied to expand 3(4x + 5) + 3 - 7x = 90 - x in example 1, leading to a simpler equation.
- โ Combining like terms simplifies 12x - 7x and 15 + 3 to 5x + 18 = 90 - x.
- ๐ The addition property of equality eliminates variables from one side by adding x on both sides.
- โ Division property of equality is the final step to solve x = 12 after using subtraction and division.
- ๐งฎ The second example involves solving 2n + 3n - 11 = 8(n - 1), using distributive and combining like terms.
- โ Subtracting 5n from both sides and simplifying the equation proves that n equals -1.
- ๐ The symmetric property is used to flip the equation n = -1 to the desired form.
- โ 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
๐ 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.
๐ 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.
๐ 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.
Mindmap
Keywords
๐กTwo-Column Algebraic Proof
๐กStatement
๐กReason
๐กDistributive Property
๐กCombining Like Terms
๐กAddition Property of Equality
๐กSubtraction Property of Equality
๐กDivision Property of Equality
๐กSymmetric Property
๐กProof
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
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