A Brief Way to Solving and Understanding Linear Equations
Many people struggle with Mathematics that it has the most despised subjects in schools and topic of discussion. It is almost rare to find students or persons who love Math, and are passionate about numbers and equations. It is even considered a headache to some that they don’t like any computations at all.
Significantly, Algebra as one of the core concepts is actually the most hated topics with all the numbers, formulas, and graphing it covers, compared to Geometry. But, do you know that it is the very foundation of our ability to solve, think, and gaze? And so, we should learn to love this concept and welcome with open mind. And, we can begin this journey with expanding our understanding about linear equations as these are fundamentals.
Defining Linear Equations
Basically, an equation is called as such if it forms a straight line on a graphical presentation. So you need to put this on a graph. It is purely dependent on constants and variables that are raised to the 1st power.
Take for example: Y = 6X + 2
This sample equation is linear due to the absence of square roots, cubes, and others.
Ax + B = 0 — This is the usual form of the linear equations. However, the places may change and it might confuse you in determining whether an equation is linear or not if the position or place is changed.
Understanding the Systems
If there are two or more equations in one item or question, of course, what do you need to do? Yes, you have to solve it. If, for an instance, you stumble upon the equation, x + y = 14; this is not part of what we call the systems of linear equations. Why? Simply because it is just one equation.
However, if ever you encounter a question or item that has 2 or more equations, like 3x +2 y =16; 7x + y= 19, then you have to solve it and you can do that by using three (3) different methods, one of which is unique and the level of difficulty is different from the other. But all of them will give you the same accurate results. So here they are:
- By using graphs
- By using the substitution method
- By using Linear combination method (Elimination)
The Graphical Solution
If we are to use graph, we should then rewrite both the equations to its slope – intercept form which is y= mx+b; where m = the slope and b = y-intercept.
3x +2 y = 16 then becomes, 2y = -3x + 16 and 7x+ y = 19 becomes, y= -7x + 19
After doing the math of this, I found out that the point where the two equations meet is at (2, 5) meaning, x = 2; y = 5. I’m pretty certain that you already know how this looks like in a graph. They meet at exactly (2, 5)
Another good example is the popular algebraic combination of -4 x -4. So what is -4x-4? If we use the graphing method, we will see that this gives us a straight line that is sloping down. The constant slope value of which is -4. If we use the formula Y = MX = B, then Y is equal to -4 x -4, M is equal to -4, and B is equal to -4. Now, what does this represent? Well, this kind of formula actually reflects the profit that anyone gets when selling something.
Another method is to find the value of x. How? We simply set the equation equal to zero. Afterwards, add -4 to both and then divide both by -4.
The Substitution Method
Based on its name, what we will do is we will substitute on and on and on. What we first have to do is to resolve one equation first (any equation would be fine)
3x+2y = 16
7x+y = 19; y = 19-7x
Then now that we know a temporary value of y we can now use it to substitute to the other equation
3x + 2(19 – 7x) = 16
3x +38 – 14x = 16
-11x = -22
x = 2 Yes! We will now then do the same thing, only different because we’ll be using x.
y + 7(2) = 19
y + 14 = 19
y = 5 So our solution set is, (2, 5)
The Linear Combination or Elimination Method
In this equation, you can either add or subtract a multiple of one of the equations from the other so that the x-terms or the y-terms will be eliminated or canceled out.
7x + y = 19; multiply the second equation to what it can be canceled and add the result to the first equation which is 3x + 2y = 16
7x + y = 19
(-2)7x + y = 19(-2)
-14 – 2y = -38
3x + 2y = 16
-11x = -22; x = 2
Then substitute 2 for x to the either of the original equations to know the value of y.
7(2) + y = 19
14 + y = 19
Y = 19 – 14
y = 5 We then arrive at the solution, (2, 5).
As you can see, linear equations are not that complicated, it is just a matter of knowing what to do in order to arrive to a specific solution or answer. In this case, we elaborated 3 methods of solving and I hope you study that for it to be muscle memory for you meaning, you won’t have a hard time thinking about it when you solve it.