### Video Transcript

In this video, we will learn how to
solve linear equations over the real numbers and represent their solutions on a
number line. We will be focusing on linear
equations primarily of the form 𝑎𝑥 plus 𝑏 equals 𝑐, where 𝑎, 𝑏, and 𝑐 are
real constants and 𝑎 is nonzero.

We recall that when solving
equations, we perform the same operations to both sides of the equation so that the
equation remains balanced. These operations will be the
inverses of the operations performed when forming the equation. As such, for equations of the form
𝑎𝑥 plus 𝑏 equals 𝑐, 𝑥 will be equal to 𝑐 minus 𝑏 divided by 𝑎. To solve the two-step equation, we
first need to isolate the 𝑥-term and then divide by its coefficient. This is equivalent to adding the
additive inverse of 𝑏 to both sides and then multiplying both sides by the
multiplicative inverse of 𝑎. We will recap this formal method in
our first example.

Find, in the set of real numbers,
the solution set of the equation two 𝑥 plus three is equal to five. Which of the following represents
the solution of the equation on a number line? Is it option (A), (B), (C), (D), or
(E)?

The first part of this question
wants us to solve a linear equation in the form 𝑎𝑥 plus 𝑏 is equal to 𝑐, where
𝑎, 𝑏, and 𝑐 are constants. In order to solve the equation two
𝑥 plus three equals five, we begin by subtracting three from both sides. And as five minus three is equal to
two, we are left with two 𝑥 equals two. Our next step is to divide through
by the coefficient of 𝑥, in this case two. As two 𝑥 divided by two is 𝑥 and
two divided by two is one, we have 𝑥 is equal to one. The solution set of the equation
two 𝑥 plus three equals five contains the solitary element one.

The second part of this question
asks us to identify the solution to the equation on a number line. Option (E) represents the solution
𝑥 is equal to eight. Option (D) is the solution 𝑥 is
equal to three. Option (C) is the solution 𝑥 is
equal to four. And option (B) represents the
solution 𝑥 is equal to two. Since the solution to our equation
was 𝑥 is equal to one, none of these options are correct. The number line that represents the
solution of the equation two 𝑥 plus three equals five is option (A). We have a solid dot representing
the single value one.

In this first example, solving the
equation was reasonably straightforward as the coefficients and constants involved
were all integers. We will now consider more
complicated equations which involve coefficients or constant terms which are
radicals or surds. These will sometimes lead to
solutions themselves which involve radicals.

Which of the following is the
solution set of the equation two 𝑥 plus two root three equals two in the real
numbers? Is it (A) one plus root three, (B)
one minus root three, (C) two plus root three, (D) two minus root three, or (E) four
minus root three?

In this question, we need to solve
a linear equation in the form 𝑎𝑥 plus 𝑏 equals 𝑐, where 𝑎, 𝑏, and 𝑐 are
constants. We solve the two-step equation two
𝑥 plus two root three equals two by firstly isolating the 𝑥-term. To do this, we subtract two root
three from both sides. This is the same as adding the
additive inverse of 𝑏 to both sides of the general equation. When we do this, the left-hand side
becomes two 𝑥. And on the right-hand side, we have
two minus two root three.

Our next step is to divide both
sides of the equation by two. This is the coefficient of 𝑥. On the left-hand side, the twos
cancel, leaving us with 𝑥. And on the right-hand side, we are
left with one minus root three. We can therefore conclude that the
solution set of the equation two 𝑥 plus two root three equals two is option (B) one
minus root three.

In our next example, the
coefficient of the unknown variable 𝑥 is a radical. As such, we will need to divide by
this radical, which will lead to an answer involving an irrational denominator. We will then need to recall the
process of rationalizing the denominator of a fraction.

Find, in the set of real numbers,
the solution set of the equation root three 𝑥 plus two is equal to five. Which of the following shows the
solution of the equation on a number line? Is it option (A), option (B),
option (C), option (D), or option (E)?

The first part of this question
involves solving a linear equation of the form 𝑎𝑥 plus 𝑏 equals 𝑐, where 𝑎, 𝑏,
and 𝑐 are nonzero constants. To solve the equation root three 𝑥
plus two is equal to five, we begin by isolating the 𝑥-term. We do this by subtracting two from
both sides. Root three 𝑥 is therefore equal to
three. Next, we divide through by root
three. On the left-hand side, the root
threes cancel and we are left with 𝑥. Our expression on the right-hand
side, three over root three, is a fraction, and its denominator is irrational. We therefore need to rationalize
the denominator by multiplying both the numerator and denominator by root three. And since root three multiplied by
root three is three, we have 𝑥 is equal to three root three over three. And this simplifies to 𝑥 is equal
to root three.

The solution set of the equation
root three 𝑥 plus two equals five is the single value root three.

The second part of this question
asks us to identify the number line that represents this solution. Since the square root of one is
equal to one and the square root of four is equal to two, we know that the square
root of three must lie between one and two. This means that we can rule out
options (B), (C), (D), and (E), as (B) has a solution between three and four, (C)
and (D) have a solution between zero and one, and (E) has solution 𝑥 is equal to
three. The correct number line is
therefore option (A). Whilst it is not required in this
question, it is worth noting that the square root of three is equal to 1.732 and so
on. And since the solid dot lies closer
to two than one, this backs up the answer of option (A).

Before looking at one final
example, we will summarize the steps involved in solving two-step linear
equations. In order to solve two-step linear
equations of the form 𝑎𝑥 plus 𝑏 equals 𝑐, where 𝑎, 𝑏, and 𝑐 are constants and
𝑎 is nonzero, we follow the following steps. Firstly, we isolate the unknown
variable by adding the additive inverse of 𝑏 to both sides of the equation. Secondly, we divide both sides of
the equation by the coefficient of the unknown variable. This is equivalent to multiplying
both sides by the multiplicative inverse of 𝑎. Finally, if this leads to a
fraction with an irrational denominator, we rationalize the denominator by
multiplying both the numerator and denominator of the fraction by the radical in the
denominator.

We will now consider one final
example where we need to follow these three steps.

Find, in the set of real numbers,
the solution set of the equation root five multiplied by root three 𝑥 minus two is
equal to four root five.

There are a couple of approaches we
could use to answer this question. For example, we could begin by
distributing the parentheses on the left-hand side. However, as the unknown variable 𝑥
appears inside the parentheses, it will be simpler to divide both sides of the
equation by root five first. When we do this, we have root five
multiplied by root three 𝑥 minus two all divided by root five is equal to four root
five over root five. On both sides of the equation, the
root fives cancel. And as such, our equation becomes
root three 𝑥 minus two is equal to four. We can then add two to both sides
of this equation to isolate the 𝑥-term. As four plus two is equal to six,
we have root three 𝑥 equals six.

Next, we divide through by the
coefficient of 𝑥, in this case root three. And 𝑥 is therefore equal to six
over root three. Since the denominator of our
fraction is a radical, we need to rationalize the denominator by multiplying the
numerator and denominator by root three. Recalling that root three
multiplied by root three is three, we have 𝑥 is equal to six root three over three,
which in turn simplifies to 𝑥 is equal to two root three. The solution set of the equation
root five multiplied by root three 𝑥 minus two is equal to four root five is the
single value two root three. We could check this answer by
substituting our value of 𝑥 back in to the original equation.

We will now finish this video by
recapping the key points. When solving equations, we always
perform the same operations to both sides. We saw that a two-step linear
equation of the form 𝑎𝑥 plus 𝑏 equals 𝑐 can be solved by first isolating the
unknown variable and then dividing by its coefficient. If we have a fractional answer with
an irrational denominator, we need to rationalize this. And we do so by multiplying both
the numerator and denominator of the fraction by the radical in the denominator. We then need to simplify this
answer. Finally, we saw that the solution
to a linear equation can be represented using a solid dot on a number line. In the case of irrational
solutions, the position of the dot will be approximate.