abMathematics

If we plot all the points that satisfy the equation y = mx + c, we’ll see that we get a line with slope m and y-intercept c. This form of the equation of the line (we’ll see another next class) is called the point-slope form equation of a line.

Now that we’re familiar with this type of equation, we can find the equation of a line with a given slope that passes through a given point.

For example, if I want to find the equation of a line with slope –2 that passes through the point A(1, 2), we know the equation will have the form y = –2x + c, but we still need to find the y-intercept (in other words, we still need to find the value of c). Here we can use the fact that the point A(1, 2) should be on our line. Since A is on our line, we should have \[\begin{align}2&=-2(1)+c\quad \text{ and so}\\ 4&=c\end{align} \]

But this means we’ve just found the “right” value for c! Now we now know that the equation for the line with slope –2, and passing through A(1, 2), is y = –2x + 4. (You can check this using GeoGebra.)

Complete the questions below before our next class.

1. Find the equation of the line with slope –1, passing through the point (1, 4).
2. Find the equation of the line with slope –1, passing through the point (2, 0).
3. Find the equation of the line that passes through (1, 3) and (2, –1).
4. Find the equation of a line perpendicular to the line you found in question 3, which also passes through the point (1, 3).

Complete the following questions before our next class.

Exercise 9D.2 questions 1–3

Now that we’ve got a definition of the slope of a line (or a line segment), we can now give easy definitions of what it means for two lines to be parallel or perpendicular.

Complete the following questions before our next class.

Exercise 9D.1 questions 1–4, 6, 10

Complete the questions below before our next class.

1. A line contains the point A(1, 3) and has a slope of 3. Find the coordinates of two other points on the same line.
2. A line contains the point A(0, 5) and has a slope of \(\displaystyle{-\frac{2}{3}}\). Find the coordinates of two other points on the same line. (Hint: associate the negative sign with the number in numerator before trying to find other points on the line.)
3. Consider the point A(1, 1). On a grid, graph a line with a slope of \(\displaystyle{\frac{3}{5}}\) that passes through point A, and a line with a slope of \(\displaystyle{-\frac{5}{3}}\) that also passes through point A. What do you notice about the two lines?