What does $b$ represent in the straight line equation?

Question

I need your help, we know that the line equation is y= mx+b but my question is what does b represent in the straight line equation and how it effect the line?

Clear example with images would be appreciated.

Thanks everybody.

Answer 1

$b$ represents the y-intercept of the line: where the line crosses the y-axis.

It can be found by setting $x = 0$ (the line crosses the y-axis when and only when $x = 0.$)

An example of such a line, $$y = 3x + \underbrace{7}_{\large b}$$ crosses the y-axis at the point $(0, 7)$.

When $x = 0$, you see that $y = 7 = b$.

Answer 2

Sometimes it's best to just look at a graph to see what's going on.

$b$ is where the line hits the $y-$axis.

The gradient (or slope) is how steep the line is.

The gradient of a line through two points $(x_1,y_1)$ and $(x_2,y_2)$ is:

$$\color{purple}{\text{gradient}=\frac{y_2-y_1}{x_2-x_1}}$$

Answer 3

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Answer 4

If the equation of a line through the origin is $y=mx$, then the line $y=mx+b$ is the same as $y-b=mx$.

If we put $y'=y-b$ then this second line is $y'=mx$, and with $y=y'+b$ we see that the original line is shifted by $b$ units in the $y$ direction.

Others have drawn the diagram. It is useful to notice, if you can, how the visual intuition is reflected in the algebra, so you can develop a more algebraic intuition too.