Linear equations

A linear equation is an equation that can be modified to the form ax + b = 0, where a≠0. A concrete example might look like this: 2x + 4 = 0. The solution to this equation is the number −2, which can probably be deduced quite logically. If there were slightly larger numbers, the deduction would not be so simple, so it would require some more concrete procedure.

The basic form of the linear equation

A linear equation is an equation that contains one unknown x, which is not multiplied, subtracted, etc. See examples of different linear equations:

$$\begin{eqnarray} 2x+7&=&0\\ x-2&=&74\\ (x+3)-(7x\cdot3)&=&-12x\\ \frac{x}{2}+\frac{3x}{72}&=&x\cdot\frac{45}{8}+\frac{x^2}{x} \end{eqnarray}$$

As you can see, a linear equation can take many different shapes. In order to solve linear equations nicely, we need to modify them to their basic form. The basic form of a linear equation looks like this:

$$ax+b=0,$$

where x is the unknown and the symbols a and b are arbitrary real numbers. The number a must not be zero, i.e. a≠0. We call the expression ax the linear term, and the expression b the absolute term.

When calculating a linear equation, we can use many different equivalent adjustments (that is, adjustments that do not change the result of the equation). It is important to know these adjustments; it is impossible to solve a more complicated equation without them. We can modify an equation to its basic form either by modifying the terms themselves that appear in the equation or we can use equivalent equation modifications.

From the previous examples: the equation 2x + 7 = 0 is already in basic form, and it is true that a = 2 and b = 7. The second equation, x − 2 = 74 is not in basic form. We use the equivalent treatment and add the number −74 to both sides of the equation (or subtract the number 74 from both sides). We get the equation x − 76 = 0. This equation is already in basic form and is valid a = 1 and b = −76.

The penultimate equation is far from the basic form, but we can modify it as well. First, we "produce" a zero on the right-hand side. We will add the expression 12x to the equation, thus zeroing out the right-hand side, because (−12x)+12x = 0. The whole equation will look like this:

$$(x+3)-(7x\cdot3)+12x=0$$

Now all we have to do is add the parentheses.

$$\begin{eqnarray} (x+3)-(7x\cdot3)+12x&=&0\\ x+3-21x+12x&=&0\\ -8x+3&=&0 \end{eqnarray}$$

This linear equation is in its basic form, and it holds that a = −8 and b = 3.

In the last equation, we have to get rid of the fractions and we also have to remove x², because there cannot be such an expression in a linear equation. However, we can simply reduce the fraction $\frac{x^2}{x}$ to just the expression x. This gives us the equation

$$\frac{x}{2}+\frac{3x}{72}=x\cdot\frac{45}{8}+x$$

We get rid of the fractions by multiplying the whole equation by 72:

$$72\cdot\frac{x}{2}+72\cdot\frac{3x}{72}=72x\cdot\frac{45}{8}+72x$$

Truncate:

$$36x+3x=9x\cdot45+72x$$

And we can convert this equation to a linear equation simply by converting all the terms to the left-hand side and adding them up:

$$-438x=0$$

It is true that a = −438 and b = 0.

How to solve a linear equation

If we have a linear equation in basic form, it is already easy to solve. Let's illustrate with an example: 3x − 18 = 0. What do we need to subtract the number 18 from to get zero? Well, from the number 18 again. So for the expression 3x − 18 to equal zero, the expression 3x must equal 18, then we get 18 − 18 = 0. When will the expression 3x equal 18? Just when does x = 6, because 3 · 6 = 18.

How do we derive the general procedure from this? The first thing we need to do is find out what the linear term ax. We do this by converting the absolute term to the right-hand side of the equation, i.e., adding −b to the equation. This gives us an equation of the form ax + b − b = −b, which is the same as ax = −b. In the case of the previous example, this would give 3x + 18 − 18 = 18, which is 3x = 18. Remember that the number b was equal to b = −18, so if we add −b, we add −(−18), and this is equal to +18.

The equation is ax = −b. How do we get x directly ? In the example we were looking for such x, which if we multiply by three we get 18. What did we actually do? We divided by 18/3 = 6. So we took the expression on the right hand side (−b) and divided it by a, in the example by three. So from ax = −b we get

$$x=\frac{-b}{a}=-\frac{b}{a}$$

The whole procedure applied to the first example looks like this:

$$\begin{eqnarray} 3x-18&=&0\quad/+18\\ 3x&=&18\quad/:3\\ x&=&6 \end{eqnarray}$$

Geometric meaning

If we modify the linear equation to its basic form, then we get a linear function on the left-hand side. The graph of the linear function is a straight line. Since the basic form of the linear equation looks like this ax + b = 0, then the solutions of this equation are all the points at which the line (the graph of the linear function) intersects the axis x (which is also the graph of the function f(x) = 0, i.e. the right-hand side).

For example, we found that the solution to 3x − 18 = 0 is x = 6. What does this mean? That the graph of the function 3x − 18 intersects the axis x at the point x = 6.

Let's try to graphically solve the equation 5x + 2 = 2x − 7. How can we proceed? We can modify the equation to its basic form, i.e. add −2x, to get 3x + 2 = −7, and then add 7: 3x + 9 = 0. Now let's plot the graph of the linear function y = 3x + 9 and see when it intersects the axis x.

The second way is to not modify the equation at all, but to have the graphs of the functions on either side of the equation drawn. That is, we plot the graphs of the functions f(x) = 5x + 2 and g(x) = 2x − 7.

These two lines intersect at one point, labeled A. What is the A x -coordinate of this point? Again, x = −3. Thus, the graphical solution of the equation 5x + 2 = 2x − 7 is the x-coordinate of the intersection of the graphs of the functions on the left and right sides.

Solved examples

Example One: Let's try to solve the equation 7x − 14 = 0. The situation is simple, the equation is in basic form, so we just apply the procedure:

$$\begin{eqnarray} 7x-14&=&0\quad/+14\\ 7x&=&14\quad/:7\\ x&=&2 \end{eqnarray}$$

Example Two: Solve the equation 2x + 10 = 0. The equation is again in its basic form, you just have to be careful that b is positive so you get a negative number on the right hand side after the adjustment:

$$\begin{eqnarray} 2x+10&=&0\quad/-10\\ 2x&=&-10\quad/:2\\ x&=&-5 \end{eqnarray}$$

Third example: solve the equation 15 − 3x = 0. The equation is almost in its basic form, we just need to get the unknown x to the first place. Then we have the form of the equation −3x + 15 = 0. Again, note that here we have a minus before the linear term, which we didn't have before. But the adjustments will be exactly the same:

$$\begin{eqnarray} -3x+15&=&0\quad/-15\\ -3x&=&-15\quad/:(-3)\\ x&=&5 \end{eqnarray}$$

We can also solve the example in a slightly different, longer way. Instead of dividing the equation directly by −3, we can first multiply the equation by −1, getting rid of the two minus signs. And then we can divide the equation by three:

$$\begin{eqnarray} -3x+15&=&0\quad/-15\\ -3x&=&-15\quad/\cdot(-1)\\ 3x&=&15\quad/:3\\ x&=&5 \end{eqnarray}$$

Example 4: Solve the linear equation

$$3+\frac{x}{4}=\frac{x}{2}$$

This equation is a bit more complicated because we have fractions. The first thing we have to do is to get rid of the fractions. It is best to multiply the whole equation by the least common multiple of all the denominators. We have two denominators, 4 and 2. The least common multiple is 4. If you don't feel like doing the math, you can multiply the whole equation by a multiple of all the denominators, i.e. 4 · 2 = 8, but that's not the recommended approach because you'll make your life one step harder. If we multiply the equation by four, we get rid of the fractions by simplifying them one by one:

$$\begin{eqnarray} 3+\frac{x}{4}&=&\frac{x}{2}\quad/\cdot4\\ 4\cdot3+4\cdot\frac{x}{4}&=&4\cdot\frac{x}{2}\\ 12+\frac{4x}{4}&=&\frac{4x}{2}\\ 12+x&=&2x \end{eqnarray}$$

Now we convert the equation to its basic form: x − 12 = 0 and from this equation we can already see that x = 12.

Fifth example / word problem: Let's have two brothers, Thomas and Jindra. Jindra went to work right after high school, while Tomas went to college, which he studied for a total of five years. Jindra received a salary of 25 000 CZK every month. Although Tomáš graduated in recreology with a specialization in reclining, he managed to get a job in the state administration, where he received CZK 35 000 per month. In total, how long will it take Tomáš to earn the same amount of money as Jindra? I.e. to clarify: in five years Jindra earned 5 · 12 · 25 000 crowns, while Tomas has nothing after five years. In how many months will this value be the same?

From the problem, we can set up a linear equation as follows: the variable x will represent the number of months we are looking for, where x = 0 denotes zero months since Tomas graduated.

The first thing we will do is to construct a function that determines Tomas's income as a function of the number of months. So, for example, if we are interested in Tomáš's income after six months, then we simply calculate that 35 000 · 6 = 210 000 crowns. In general, we can write that after x months, Tomas earned a total of 35 000 · x crowns.

For Jindra it would be similar: 25 000 · x, but we have to add the amount he earned while Tomas was studying. Tomas studied for five years, so Jindra earned 5 · 12 · 25 000 crowns during his studies. The whole prescription for Tomas would be: 5 · 12 · 25 000 + 25 000 · x.

We put these functions into the equation because we are interested in for which x these earnings are equal:

$$ 5 \cdot 12 \cdot 25 000 + 25 000 \cdot x = 35 000 \cdot x $$

We simply modify the equation by transferring all x to one side of the equation:

$$ 10000x = 5 \cdot 12 \cdot 25 000 $$

And finally, we just divide 10 000:

$$ x = 5 \cdot 12 \cdot 2{,}5 = 150 $$

In 150 months, they will both have the same money. Jindra earned a nice 5 · 12 · 25 000 = 1 500 000 crowns during Thomas's time as a student, plus in the next 150 months he earned 25 000 · 150 = 3 750 000, for a total of 1 500 000 + 3 750 000 = 5 250 000 crowns. Tomas only worked these 150 months and earned 35 000 · 150 = 5 250 000 crowns in that time. We can see that he really earned the same after five years of study and another 150 months.

Other examples

Calculate the following linear equation:

$$2\cdot(x - 7) = 6$$

First multiply the parenthesis:

$$2x-14=6$$

Now we keep the expression with the unknown on the left side and transfer everything else to the right side:

$$2x=20$$

and finally divide by two.

$$x=10$$

Calculate the equation

$$3\cdot(3 - x) + 5\cdot(x - 2) = 0$$

Again, multiply the parentheses first:

$$9-3x+5x-10=0$$

Add up what can be added and transfer the variables to the left side and the rest to the right side:

$$\begin{eqnarray} 9-3x+5x-10&=&0\\ 2x-1&=&0\\ 2x&=&1 \end{eqnarray}$$

Finally, divide the whole equation by two:

$$x=\frac12$$