Straight Lines Part 1 (y = mx + c)

Understanding Straight Line Equations


We know how to plot coordinates, and thereby plot the inputs and outputs of equations and functions. If you don't check out the plotting straight lines and curves page first and then read this page. IT IS NOT HARD.
The equation of a straight line is written in the form of y = mx + c
y is the output.x is the input.m is the letter that you may have not come across, and this represents the gradient/steepness of the line.c is the y-intercept of the line.
An example of a straight line equation is y = 2x - 5, in this equation, m = 2, c = -5.
You know to plot points on a graph, if you wanted to plot the straight line y = 2x - 5, for when x is equal to -3 to 8, you would know how to do this.
But lets understand further what the gradient and y-intercept of a line is, using the example y = 2x - 5.
Lets draw y = 2x - 5, for when x is equal to -1 to 3.
so
when x = -1, y = 2(-1) - 5 = -2 - 5 = -7when x = 0, y = 2(0) - 5 = 0 - 5 = -5when x =1, y = 2(1) - 5 = 2 - 5 = -3when x = 2, y = 2(2) - 5 = 4 - 5 = -1when x = 3, y = 2(3) - 5 = 6 - 5 = 1
So the coordinates we need to plot are
(-1,-7), (0,-5), (1,-3), (2,-1), (3,1)
This is shown below in figure 1.

Figure 1

c is the Y-Intercept


You can see that the line crosses the y axis at the point (0,-5). The value on the y axis at which the line crosses is known as the y intercept. Remember, we have drawn y = 2x - 5, this is an equation of a straight line, as it is in the form of y = mx + c. In our case, c = -5.
Lets think about it logically as well. We know that if we wanted to find the point at which a straight line crosses the y axis, it would occur when x = 0. So when we substitute x = 0 in any straight line equation we would get the y-intercept, as shown below.
When x = 0, y = 2(0) - 5 = 0 - 5 = -5in coordinate form we can write (0,-5), and this point is the point at which the line crosses the y axis.
If we had the equation y = 3x + 2, the y-intercept would be 2 because the line crosses the y axis at (0,2), here x = 0 and y = 2.

m is the Gradient


The gradient is the steepness of the curve.
In our straight line equation y = 2x - 5, m is equal to 2. So lets understand what this 2 actually means.
Remember that when we plot equations we are plotting the inputs and outputs. So look back at figure 1 at the point (-1,-7), this tells us that when x (the input) is equal to -1, y (the output) is equal to -7. If we were to increase x by 1, the output should increase by m (the gradient) which is 2. So lets increase x by 1, so that x = 0.
What does y equal?
When x = 0, y = -5. Has the output increased by 2?
Yes.
SO the gradient tells us if we were to increase x by 1, the output (y) value would increase by the value of m. It also thus tells us if we were to decrease x by 1, the output (y) value would decrease by the value of m.

LETS THINK ABOUT THIS ALGEBRAICALLY


We have the straight line equation y = mx + c
If we were to increase x by 1, we can write
y ≠ m(x + 1) + c (here we have increased x by 1, by adding 1 to the x.But this is wrong as we changed one side without changing the other, that is why we have used the IS NOT EQUAL TO SIGN (≠). But what have we actually done?m(x + 1) + c = mx + m + cWe have actually just added m, and this shown once we expand the bracket.So we must add m to the other side, which is y, so that both sides remain balanced (remember what we do to one side we must do to the other).
So add m to y
y + m = m(x + 1) + c
So this tells us that when we increase the x (the input value) by 1, the output would increase by m which is the gradient.
If we were to decrease x by 1, we can write
y ≠ m(x - 1) + c (here we have decreased x by 1, by subtracting 1 from the x.But this is wrong as we changed one side without changing the other, that is why we have used the IS NOT EQUAL TO SIGN (≠). But what have we actually done?m(x - 1) + c = mx - m + cWe have actually just subtracted m, and this shown once we expand the bracket.So we must subtract m from the other side, which is y, so that both sides remain balanced (remember what we do to one side we must do to the other).
So subtract m from y
y - m = m(x + 1) + c
So this tells us that when we decrease the x (the input value) by 1, the output would decrease by m which is the gradient.