Finding the Equation of Line

Think of an equation like $y=3x+7$y=3x+7 as an expression that performs a specific function. Any value of $x$x considered will be first tripled to $3x$3x and then that new value $3x$3x will be increased in size by $7$7. We call the result $y$y. So for $x=4$x=4, $y=19$y=19. In particular for $x=0$x=0, $y=7$y=7, and this is why the line associated with $y=3x+7$y=3x+7 has a $y$y-axis intercept of $7$7.

The line itself is straight - it has a constant slope. This may not be immediately obvious, but one way we can show this is as follows:

We know that $\left(0,7\right)$(0,7) satisfies $y=3x+7$y=3x+7. Now imagine the set of all points $\left(x,y\right)$(x,y) so that the slope between them and $\left(0,7\right)$(0,7) is fixed at the value $3$3.

Thus we can write $\frac{y-7}{x-0}=3$y−7x−0​=3.  This means that $y-7=3x$y−7=3x or that $y=3x+7$y=3x+7. This is precisely the equation we have been discussing, and so the line is straight with a constant slope of $3$3. 

More generally the equation of any straight line passing through $\left(x_1,y_1\right)$(x1​,y1​) with slope $m$m must be determined by setting $\frac{y-y_1}{x-x_1}=m$y−y1​x−x1​​=m.

By multiplying both sides of this equation by the denominator $\left(x-x_1\right)$(x−x1​) we have the point slope formula for a straight line, given as:

$y-y_1=m\left(x-x_1\right)$y−y1​=m(x−x1​)

So, for example,  the line that passes through $\left(2,3\right)$(2,3) with slope $m=5$m=5  must have the equation given by $y-3=5\left(x-2\right)$y−3=5(x−2). Simplifying this we see that $y=5x-7$y=5x−7. 

Lets look at two examples:

Question 1

The line with slope $m=-\frac{2}{5}$m=−25​ and $y$y - intercept $-3$−3 is given by $y=-\frac{2}{5}x-3$y=−25​x−3

Question 2

The line passing through $\left(-2,5\right)$(−2,5) and $\left(6,-1\right)$(6,−1) has a slope given by $m=\frac{5-\left(-1\right)}{-2-6}=-\frac{3}{4}$m=5−(−1)−2−6​=−34​ and so, using the point $\left(-2,5\right)$(−2,5) ( either of the two points can be used) we can write the equation down as $y-5=-\frac{3}{4}\left(x-\left(-2\right)\right)$y−5=−34​(x−(−2))

In steps, we can simplify this as follows:

$4y-20=-3x-6$4y−20=−3x−6

$3x+4y-14=0$3x+4y−14=0

    The graph is shown here, with the x and y intercepts indicated:

         

      

Worked Examples

Question 3

Question 4

State the slope and y-intercept of the line with equation $y=-1-\frac{8x}{9}$y=−1−8x9​

Question 5

Question 6