Video Transcript
In this video, we will learn how to
write and solve one-step addition and subtraction equations, paying close attention
to the addition and subtraction property of equality. Before we start writing and solving
equations, let’s think about what we know about an equation.
An equation is a mathematical
statement that says something is equal to something else. For example, we could say that five
is equal to five. If we were to take this positive
three block and put it on the left side of this equation, all of a sudden the scale
would be unbalanced. You would have eight on the left
and only five on the right. But if we took a positive three
block and put it on the right, the scale comes back into balance as both sides are
equal to eight. This illustrates the addition
property of equality. It tells us if you add the same
value to both sides of an equation, the sides remain equal. This also applies to
subtraction. Going back to our five equals five,
if we subtract three from both sides of the equation, we end up with two on either
side and the equation is balanced. And so, we can say If you subtract
the same value from both sides of an equation, the sides remain equal.
Let’s consider one other property:
the inverse property of addition. This tells us that any number added
to its opposite will equal zero. The opposite of negative three is
positive three. That means if we add three to both
sides of the equation, on the left, we would have five plus zero, which in the end
is just equal to five. And the same thing would happen on
the right. Five plus zero equals five. All three of these properties are
tools that we’ll need to help us solve one-step equations.
But what does that mean, “solve the
equation”? If we have an equation like this,
𝑥 plus three equals five, we have an unknown value that’s being represented by a
variable 𝑥. To solve the equation, we find the
unknown value that makes the statement true. If we’re wondering what plus three
equals five, we know that two plus three equals five. And so, we say that 𝑥 equals
two. Here, we would call two the
solution of the equation. The solution is the unknown value
that makes the statement true. And that means a question might ask
you to solve the equation. Or it might ask, what is the
solution of the equation?
So let’s consider an example.
Given that five plus 𝑛 equals
negative five, find the value of 𝑛.
We have the equation five plus 𝑛
equals negative five. This is telling us that when we add
some value to five, the result is negative five. To find out what 𝑛 is, we need to
think about the inverse property of addition. The inverse property of addition
says that any number added to its opposite will equal zero. Since five and 𝑛 are being added
together, if we subtract five, then five minus five will equal zero. But we also know the subtraction
property of equality, which tells us that if we subtract five from one side of the
equation, we must subtract five from the other side of the equation to keep it
balanced. If five minus five is zero, then on
the left side of the equal sign, we only have the variable 𝑛. And on the right side, we have
negative five minus five, which equals negative 10. And so, we found that 𝑛 equals
negative 10.
If you start with five and add
negative 10 to that five, you’ll get negative five. This is a vertical method for
solving the problem. We do this by writing the next line
directly under the line above it. The other way to do it would be to
use a horizontal method. We began with five plus 𝑛 equals
negative five. But then, in the next line, we’ll
add the negative five in the row with our original equation. So that it says five minus five
plus 𝑛 equals negative five minus five. Both methods show that 𝑛 is equal
to negative 10.
Here’s another example.
Solve for 𝑥: 𝑥 plus 4.8 equals
negative 8.9.
This equation tells us that some
value when we add 4.8 is equal to negative 8.9. When we see the word “solve” here,
it means that we want to know which value for 𝑥 makes this statement true. If the equation has add 4.8, to
solve it, we’ll need to do the inverse of adding 4.8. The inverse property says that any
number added to its opposite will equal zero. But we also know, based on the
subtraction property of equality, that if we subtract 4.8 from the left side of the
equation, we must subtract 4.8 from the right side of the equation to make sure they
stay equal. We now have on the left 𝑥 plus
zero. We can just write that as 𝑥. And then, we have negative 8.9
minus 4.8, which equals negative 13.7. If we’ve calculated this correctly,
we can plug in negative 13.7 in for 𝑥. And we’ll get a true statement.
We need to know is negative 13.7
plus 4.8 equal to negative 8.9. If we want to add 4.8 to negative
13.7, we subtract the two values first, which gives us 8.9 and then it takes the
sign of the larger number. 13.7 was negative. So, the final answer will be
negative. Negative 13.7 plus 4.8 does equal
negative 8.9, which means the solution of 𝑥 is negative 13.7.
For the next example, we’ll have to
first write an equation and then solve it.
William is playing a board
game. From start, he moves 10 spaces
forward. In his next turn, he moves six
spaces back. How many spaces away from start is
he now?
If William begins on start and he
moves 10 spaces forward. On his next turn, he moves six
spaces back. We want to know how many spaces is
he away from start. For this unknown value, we can use
the variable 𝑥. What kind of equation can we write
to model the situation? We can write it a few different
ways. First, William went forward 10
spaces. So, we can start with positive
10. And then, he went back six
spaces. We can represent that
mathematically with negative six. If you take positive 10 and
subtract six, you’ll get 𝑥, the number of spaces away he is from the start. 10 minus six equals four. And that means our 𝑥-value is
four.
Currently, the way it’s written: it
says four equals 𝑥. But it’s fine to rearrange it in a
more common way: 𝑥 equals four. 10 minus six equals 𝑥 is only one
way to model this situation with an equation. We could say that 𝑥 — the number
of spaces away from start William is — plus the six places backwards he walked must
be equal to the 10 total forward spaces. If 𝑥 plus six equals 10, then we
can solve the problem by subtracting six from both sides of the equation. 𝑥 plus six minus six equals 𝑥
plus zero, which is 𝑥, and 10 minus six equals four. Both methods show us that William
is four places away from start.
Here’s another word problem
example.
In 2016, Mississippi and Georgia
had a total of 21 electoral votes. If Mississippi had six electoral
votes, solve the equation six plus 𝑔 equals 21 to find the number of electoral
votes that Georgia had 𝑔.
Here, we’ve actually been given an
equation: six plus 𝑔 equals 21. And our goal is to solve for
𝑔. To solve for 𝑔, we want to get the
𝑔 by itself. We want to isolate this 𝑔
variable. To do that, we’ll need the inverse
property of addition, which tells us that any value if you add its opposite equal
zero. To get 𝑔 by itself, we want to get
rid of the six. And we can do that by subtracting
six. Six minus six equals zero. But we also know that to keep both
sides of this equation equal, if we subtract six from one side, we must subtract six
from the other side. On the left, it would be zero plus
𝑔. Or we can write that simply as
𝑔. And on the right, 21 minus six
equals 15. And so, we found that 𝑔 equals
15. It’s worth checking to make sure
that that’s true. Is six plus 15 equal to 21? Yes, it is. This tells us that 𝑔 is in fact
15. So, Georgia had 15 votes and
Mississippi had six votes.
We’ll consider one final
example.
A diver began his ascent to the
surface. He ascended 20 meters to his next
decompression stop and must ascend another 32 meters to return to the surface. Write and solve a subtraction
equation to find the diver’s original depth before he started ascending.
The diver started at a depth that
we don’t know. We can call this step 𝑥. We do know that the diver ascended
20 meters and still needs to ascend 32 meters before the diver reaches the
surface. We want to write and solve a
subtraction equation to find 𝑥, the diver’s original depth. If we began with 𝑥, the diver is
𝑥 meters deep and he rises 20 meters. Mathematically, we need to
represent that with subtraction because he is not as deep as he was before. Even though he’s going up, he is
becoming less deep. It’s his original depth minus the
20 meters he rose. And 𝑥 minus the 20-meter ascent
will equal the 32 meters that remain, which means one equation is 𝑥 minus 20 equals
32.
To solve this equation, we add 20
to both sides. 32 plus 20 equals 52. And so, we can say that the diver’s
original depth was 52 meters. We’ve written a subtraction
problem. And we’ve solved for the missing
depth to show that the diver’s original depth before he started ascending was 52
meters.
We’ll now think about the key
points that are needed to solve one-step equations with addition and
subtraction. Solving equations is finding the
value or values of the variable that make the equation true. These values that make the equation
true are called solutions. To solve equations, we use the
addition and subtraction property of equality, which tells us if we add or subtract
something from one side of the equation, we must do the same thing to the other side
of the equation to make them remain true. And finally, we know the inverse
property of addition, which tells us if we add the opposite value to a value, the
result is zero. Using these key points, you can
write and solve one-step equations with addition and subtraction.
