Linear equations

You can find Linear equations in different disciplines, such as cost of production, the relation between temperature scales, depreciation, and even simple interest. The most important thing about linear equations is their simplicity, which simplifies their solution and makes it possible to get a direct interpretation of the results. In this topic, you will learn how linear equations appear in common situations, how to solve them, and will face a financial problem of simple interest.

Linear equations in real life

If you are making your favorite cake and double the portion of all the ingredients, you will be able to bake two cakes. This means that there is a linear relation between the ingredients and the cake.

However, there is not always a linear relation in every situation. For example, the relation between your actual learning and the time you spend studying is not linear. If you study twice as long as you usually do, that does not mean that you will learn twice as much, simply because as time goes by, you get tired and slightly decrease your retention capacity.

Mainly in physics and chemistry, the Kelvin temperature scale is used because it describes temperatures in an absolute way. Despite this, in everyday life, it is more common to use the Celsius scale. In this case, a physicist who takes temperature measurements with a device using Celsius (C) degrees would need to convert their measurements to Kelvin (K) using the linear equation $K = C + 273.15$.

Solve linear equations

Linear equations look like this:$$ax + b =0$$This means that complex expressions like $x^2, \cos(x), \log(x)$, etc. do not appear. Solving a linear equation means isolating $x$. For this purpose, what you must remember is that the equal sign represents an equilibrium, which means that what is on the left side is exactly the same as what is on the right side. So if we do something on one side of the equation, we have to do it on the other side. This is why when we solve a linear equation, we can:

• Add or subtract the same amount from each side
• Multiply or divide both sides by the same amount

Then, to solve linear equations, we simply have to isolate $x$, for which we first get rid of $b$ by subtracting it from both sides of the equality:$$ax = -b$$And then we divide both sides by $a$:$$x = \frac{-b}{a}$$So the solution to $ax+b=0$ is $x = \frac{-b}{a}$!

In applications, it is common that we do not start with the equation $ax +b = 0$, but a common strategy is to manipulate the equation so that we isolate $x$ on one side. For example, if we had$$2x + 5 = 3x - 4$$The first thing we would do is to subtract $3x$ from both sides to ensure that $x$ is now only on one side:

$$-x + 5 = -4$$We see that we can then subtract $5$ from both sides to leave $x$ alone:

$$-x = -9$$Finally, we multiply by $-1$ both sides and then $x = 9$.

Example: simple interest

Under simple interest, an initial investment $P$ with a rate of return $i$, after $t$ years, generates an accumulated amount $A$ equal to:$$A = P(1+it)$$Depending on the type of question, the unknown could be $A, P, i$, or $t$. In order to solve the problem, you would first have to know the value of the other variables.

Suppose you want to buy your dream car, whose price is $$25,000$. Today you only have $$16,000$, so you decide to invest your money with the intention that in a few years you will be able to buy the car. At your favorite bank, they give you a rate of return of $5\%$. If the interest is simple, in how many years will you be able to buy the car?

We know quite a lot of things, like that the money you have today is the initial investment, that the rate of return is $5\%$ and that the amount you would like to accumulate is the value of the car. We can substitute these values into the simple interest formula:

$$25000 = 16000 (1+0.05t)$$Clearly, what you now need to know is the time $t$ it will take you to save to buy the car, so this is the unknown variable.

We can start by dividing both sides by $\$16,000$:$$\frac{25000}{16000} = 1+0.05t$$After simplifying and subtracting $1$ from both sides:

$$\frac{25}{16} - 1 =0.05t$$Finally, we divide by $0.05$:$$t = \frac{\frac{25}{16} - 1}{0.05} = \frac{1}{0.05}\left(\frac{25}{16} - \frac{16}{16}\right) = \frac{100}{5}\frac{9}{16} = \frac{900}{80} = \frac{90}{8} = 11.25$$This means that within $11$ years and $3$ months (a quarter of a year) your initial investment of $\$16,000$ would grow to $\$25,000$.

Conclusion

• Linear equations look like $ax +b = 0$.
• When we solve a linear equation, there is always a unique solution $x = \frac{-b}{a}$.
• As in any equality, the important thing is to maintain balance: if we add or subtract a quantity from one side, we must do the same on the other side. The same goes for multiplication and division.