Video Transcript
In this video, we will learn how to
solve two-step equations. To do that, we will use the balance
method. That’s a general method for solving
equations based on the additive and multiplicative properties of equality.
Before we try to solve this
equation, let’s think about what it’s doing. This equation is giving us
information about an unknown value. Here, that unknown value is
represented with the letter 𝑥. The simple equation tells us that
when this 𝑥-value has undergone some operations, it led to the result of 19. To solve this equation. We want to reverse all the
operations that this 𝑥 has gone through. And we want to do it in the
contrary order. That is, the opposite order that it
happened in.
Our first term two 𝑥 tells us that
some value is doubled. And whatever that amount was, you
add five to it. And the result was 19. Two 𝑥 plus five equals 19. We can visualize this with a bar
diagram. Two 𝑥 plus five is equal to
19. The bar diagram suggests that we
could divide 19 into two pieces. If we took a piece that was five,
the remaining portion on the left would be equal to 19 minus five, which is 14
because 14 plus five equals 19.
If we take away the part of the bar
that represents five from the top bar and the bottom bar, the two bars are still
equal. We have a bar that’s two 𝑥 in the
top and a bar that equals 14 in the bottom. What if we divide both of these
bars in half. Two 𝑥 divided by two equals 𝑥,
and 14 divided by two equals seven. If two 𝑥 equals 14, then one 𝑥
equals seven. Going back to this statement, some
value is doubled, then we add five, and it equals 19. When we double seven, we multiply
it by two. Two times seven is 14, plus five
equals 19.
To help us think about how to solve
this mathematically, let’s visit the idea of a balance. On the left, we have two 𝑥 plus
five. And on the right, we have 19. But inside that 19, there’s also a
five. We know that 14 plus five equals
19. On the right then, we can switch
out the 19 for one block of 14 and one block of five. If we take away five from both
sides, we are subtracting five from both sides of our equation. And what will be left? On the left side, we’ll have two
𝑥. And on the right side, we’ll have
14.
If we divide two 𝑥 by two, we get
𝑥. And if we divide 14 by two, we get
seven. Which tells us that 𝑥 equals
seven. We have subtracted by five and
divided by two, which is working in reverse order of the equation we were given. The equation we were given is
multiplied by two and then add five. But why is this possible? Why does the balance method
work? It seems really intuitive. But there are two main properties
that this is showing. They’re properties of equality for
multiplication and addition.
In the multiplication property of
equality, we know that if both sides of the equation are multiplied by the same
number or if both sides of the equation are divided by the same number, then the
equality is still true. We might write it in symbols like
this. If 𝑎 equals 𝑏, then 𝑎 times 𝑐
is equal to 𝑏 times 𝑐. Similarly, if the same amount is
added to or subtracted from both sides of the equation, then the equality is still
true. And written in symbols, if 𝑎
equals 𝑏, then 𝑎 plus 𝑐 equals 𝑏 plus 𝑐.
Let’s put these properties into
action by looking at some examples.
A number is tripled and seven is
subtracted from the result. If the answer is 17, what is the
number?
The first thing we need to do here
is take this statement and turn it into a mathematical one. We have an unknown number. And to represent an unknown number,
we need a variable. It’s common to use the variable
𝑥. And our number is tripled. If we triple 𝑥, it becomes three
𝑥. Whatever this three 𝑥 is, seven is
subtracted from that. Because we don’t know what three 𝑥
is, we have to write three 𝑥 minus seven. And the answer is 17. So, we can say that three 𝑥 minus
seven equals 17.
If we start with this equation,
three 𝑥 minus seven equals 17, we need to work in the reverse order. The last thing we did to our number
was subtract seven. In order to reverse this operation,
we need to add seven. But in order to keep both sides of
the equation equal, we need to add seven to both sides based on the addition
property of equality. On the left, three 𝑥 minus seven
plus seven equals three 𝑥. And on the right, 17 plus seven
equals 24.
To undo the subtract seven, we
added seven to both sides. How would we then undo this tripled
value? If we remember that triple means
times three, to undo that, we would divide by three. We need to then divide by three on
both sides. Three 𝑥 divided by three equals
𝑥. 24 divided by three equals
eight. We can do this because of the
multiplication property of equality. Which says if we multiply or divide
by the same thing on both sides of the equation, the statement remains
equivalent. Using these two properties, we’ve
shown that 𝑥 equals eight.
In problems like this, it can be
worth it to check and make sure that’s true. If our number is eight, and it’s
tripled, eight times three equals 24. Seven is subtracted from that
result. 24 minus seven should equal 17. And it does. So, we’ve correctly identified the
missing number, which is eight.
Here’s another example.
Find the solution set of seven 𝑥
plus 11 equals negative 24 in the set of all integers.
We remember that this symbol that
looks a little bit like a Z is the set of all integers. We know that seven 𝑥 plus 11
equals negative 24. And the solution set will be the
values for 𝑥 that makes seven times 𝑥 plus 11 equal to negative 24. In order to solve for 𝑥, we need
to get 𝑥 by itself. And the first thing we can do is
subtract 11 from both sides of the equation. If we subtract 11 from both sides,
it keeps this equation balanced.
Seven 𝑥 plus 11 minus 11 equals
seven 𝑥. And negative 24 minus 11 equals
negative 35. And now, we have seven 𝑥 equals
negative 35. It means seven times some number
equals negative 35. In order to undo that multiplied by
seven, we need to divide both sides of this equation by seven. Seven 𝑥 divided by seven equals
𝑥. And negative 35 divided by seven
equals negative five. This means that when we multiply
seven by negative five and then add 11, we should get negative 24.
Seven times negative five is
negative 35. Is negative 35 plus 11 equal to
negative 24? It is. And so, we found the value that 𝑥
must be is negative five. We should be careful here because
our question is asking us for a solution set. And that means we’ll need to use
the curly brackets. But the only integer, the only
value, for 𝑥 that makes this statement true is negative five. And that means that’s the only
value that goes into the solution set. The solution set of this equation
is negative five.
Here’s another example.
Find the solution set of the
equation negative four 𝑥 plus three equals four.
We’re given the equation negative
four 𝑥 plus three equals four. This negative four 𝑥 means
negative four times 𝑥. And whatever that value is, we’re
adding three to it. We find the 𝑥-value by isolating
the 𝑥, by getting it by itself on one side of the equation. In order to get 𝑥 by itself, we
first subtract three from both sides. Negative four 𝑥 plus three minus
three equals negative four 𝑥. And four minus three equals
one.
Because we now have negative four
times 𝑥 equals one, to find 𝑥 by itself, we need to divide both sides of the
equation by negative four, which you might write like this or in the fraction form
like this. Negative four 𝑥 divided by
negative four equals 𝑥. And we need to be careful on the
right side because we have one divided by negative four. We can write that as a fraction,
negative one-fourth, or in the decimal format, negative 0.25.
Once we found negative one-fourth
for 𝑥, it’s worth plugging it back in to check. Negative four times negative
one-fourth equals one. And one plus three does equal four,
which means we found a correct value for 𝑥. But we should be careful here
because the question is asking for a solution set. And that means we’ll use the curly
brackets. There’s only one value in the
solution set. And that’s negative one-fourth,
which in set notation looks like this.
In the next example, we have 𝑥
being divided by something instead of multiplied by something.
Find the solution set of 𝑥 over
four minus three equals negative five in the set of all integers.
First, we remember that this symbol
is the set of all integers. And then, we can turn our attention
to solving for 𝑥. If 𝑥 over four minus three equals
negative five, we want to try to isolate 𝑥, to get 𝑥 by itself. And the first thing we should do in
order to do that is add three to both sides. Once we do that, we’ll get 𝑥 over
four is equal to negative two because negative five plus three equals negative
two.
We know that 𝑥 over four means 𝑥
divided by four. Some value divided by four equals
negative two. To find out what that value is, we
need to do the opposite. If 𝑥 is being divided by four, we
need to multiply both sides of the equation by four, like this. Four times 𝑥 over four equals 𝑥,
and negative two times four equals negative eight. At this point, we should check and
see if that’s true.
Is negative eight divided by four
minus three equal to negative five? Negative eight divided by four is
negative two. And negative two minus three is
negative five. This means that 𝑥 is equal to
negative eight. But here our solution needs to be
given as a set notation. That’s the curly brackets. And the solution set for this
equation is just negative eight.
In all of our previous examples,
the first thing that we’ve done is add or subtracted a value. It’s not always the case that we do
addition or subtraction first. Let’s look at this example and see
why that might not be the case.
Solve 𝑥 plus three over five
equals 10.
We know that 𝑥 plus three over
five equals 10. And to solve for 𝑥, we need to get
it by itself. We need to isolate it. And that means we need to undo the
operations that are being done to 𝑥. In the numerator, we have some
number 𝑥 plus three. And whatever that number plus three
is is being divided by five. And that means to find out what
that number is, we first need to undo this division.
And we can do that by multiplying
by five. But if we multiply by five on the
left, we need to multiply by five on the right. On the left, the five in the
numerator and the five in the denominator cancel out, leaving us with 𝑥 plus
three. And on the right, we have 10 times
five, which is 50. And now, we have a simple equation,
𝑥 plus three equals 50. So, we subtract three from both
sides. And we get 𝑥 equals 47.
Why couldn’t we subtract three from
both sides? What would happen if we did? On the left, we would have 𝑥 plus
three over five minus three. And on the right, we would have
seven. Now, this is still a true
statement. 𝑥 plus three over five minus three
does equal seven. However, we’re no closer to solving
for 𝑥. And that’s because the entire
numerator is being divided by five. And we can’t take this plus three
out of the numerator. This is because we have to reverse
the operations in the contrary order than when we started.
Let’s do a quick recap on the key
points for solving two-step equations. The balance method is a general
method for solving equations based on the additive and multiplicative properties of
equality. These properties state the
following. If the same amount is added to or
subtracted from both sides of the equation, then the equality is still true. And if both sides of the equation
are multiplied by the same number or if both sides are divided by the same number,
then the equality is still true. Based on this, we say that a simple
equation can be solved by reversing all the operations in the contrary order to
yield the missing value.
