During class on Monday, April 8th, you should read the examples in Section 3B of the textbook, then complete the questions listed below.
Exercise 3B questions 1–4
9 Extended: Algebraic Expansion
Complete the following questions before our next class.
Exercise 3A.1 questions 1–9 (3 each)
Exercise 3A.2 questions 1–3 (3 each)
Exercise 3A.3 questions 1–2 (3 each)
Exercise 3A.4 questions 1–2 (3 each)
9 Extended: Elimination
Complete the following questions before our next class.
Exercise 16D questions 1 and 2
9 Extended: Simultaneous Equations Part 1
Complete the following questions before our next class.
Exercise 16A.1 question 2b
Exercise 16A.2 questions 1b, 2a
Exercise 16B questions 1def, 2abc (these are similar to what we did in class—substitute in one equation to get one equation with a single variable, solve for that variable, then use one of the original equations to solve for the remaining variable)
Exercise 16C questions 1abc, 2def, 3
9 Extended: Term Test
On Wednesday, February 13th we’ll have a test on all of the topics we’ve covered so far this year. Those topics are listed below.
- Number Sets (\(\mathbb{N}, \mathbb{R}\), etc.)
- Number lines (representing 1 ≤ x < 4, etc.)
- Operations with sets (\(A \cup B\), etc.)
- Radicals (simplifying, adding, multiplying, rationalizing the denominator, etc.)
- Trigonometry (finding sides, finding angles, working in 2D or 3D)
- Coordinate Geometry (finding the slope, finding midpoints, parallel and perpendicular lines, equations of lines)
- Word problems involving any of the above
In order to prepare for the test, the following questions may be helpful. We’ll discuss answers to these questions on Thursday. (Note that you will probably have done some of these before—make sure you still remember how to do them, and if you are absolutely sure you know how to solve a question, skip it and look for a harder question to try!)
Review Set 2B
Review Set 4A
Review Set 13A
Review Set 9A
9 Extended: General Form Equation of a Line
If we are given a point on a line, and the slope of that line, we already know how to find the equation of that line in slope-intercept form. How can we find the general form equation of that same line?
In one sense it’s very easy, we can just move the x-term to the same side as the y-term. For example, the line with slope-intercept form \[\begin{align}y&=3x+1 \quad\textrm{can also be written as}\\y-3x&=1\end{align}\]
which is in general form.
There is, however, another easy way to find either the general form equation of a line that you may also want to use. If you want to learn that method, have a look at page 174 in our textbook. You can use any method you like!
Complete the following questions before our next class.
Exercise 9E.1 questions 1–3, 4ef, 5, and 7 (if you want a challenge, try 11 as well!).
9 Extended: Equations of Lines
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.
- Find the equation of the line with slope –1, passing through the point (1, 4).
- Find the equation of the line with slope –1, passing through the point (2, 0).
- Find the equation of the line that passes through (1, 3) and (2, –1).
- Find the equation of a line perpendicular to the line you found in question 3, which also passes through the point (1, 3).
9 Extended: Collinear Points
Complete the following questions before our next class.
Exercise 9D.2 questions 1–3
9 Extended: Parallel and Perpendicular Lines
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
9 Extended: Working with the Slope
Complete the questions below before our next class.
- 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.
- 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.)
- 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?