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A straight line is perhaps the simplest graph. What kind of equation in x and y would produce a straight line for its graph? Suppose we are given two distinct points Q(a,b) and R(c,d). There is one and only one possible line that passes through the points Q and R (see figure below).
If P(x,y) is on the line, then the two triangles DQRS and DQPT are similar, and therefore we must have: . Using the coordinates of the points P, Q, R, S, and T, this means: . Cross-multiplying, we get: . Simplifying this last equation, we get: . If we let A = b - d, B = c - a, and C = , then the equation becomes . Note that the numbers A, B, and C are completely determined by the points Q and R.
If the point P(x,y) is not on the line, the triangles DQRS and DQPT are not similar and therefore the ordered pair (x,y) would not be a solution of the equation . In other words, P(x,y) is on the line if and only if (x,y) is a solution of the equation . That is, the equation of the line is: .
Example 1 if the line is given by the points Q(1,3) and R(2,5), then the equation of the line is . Upon simplification, we get . The equation of the line passing through the points Q(1,3) and R(2,5) is: .
Equation of a line: standard form
In summary, a line is the graph of an equation of the form , where A, B, and C are constant real numbers.
Slope of a line
Refer back to Figure 1. Regardless of where the point P(x,y) is on the line, the triangles DQPT and DQRS are similar and as such, we noted that . Notice that the right hand side of this equality does not depend on P and therefore is constant. In other words, regardless of where the point P is on the line, the ratio is always the same (constant). Such a constant is referred to as the "slope" of the line and is usually denoted by the letter m. Moreover, RS is referred to as the "rise" and QS as the "run" (see depiction below):
Example 2 (i) Find the slope of the line passing through the points Q(-3,2) and R(3,5). (ii) Find the slope of the line passing through the points Q(1,1) and R(3,-3). (ii)
Solution (i) We know that the slope = . All we have to do is calculate the rise and the run and then take their ratio. The rise is the difference in the y-coordinates (ordinates) of the two points: 5 - 2 = 3. The run is the difference in the x-coordinates (abscissas) of the two points: 3 - (-3) = 6. So the slope is .
(ii) The rise is 1 - (-3) = 4. The run is 1 - 3 = -2. So the slope is 4/(-2) = -(1/2). Note 1 When calculating the rise and the run, the order of the points should be the same. That is, if we calculate the rise as the ordinate of R minus the ordinate of Q, then the run should be the abscissa of R minus the abscissa of Q (or vice versa).
Example 3 Consider the line whose equation is given by . (i) Find five points on this line. (ii) Calculate the ratio: rise/run for each pair of the points you found in (i). What do you observe? What does this confirm in the earlier discussion?
Solution (i) As you recall, we find points on the graph of an equation by setting one of the two variables x or y to a specific value and solving for the other variable. If we set x to 0, we get for y. This tells us that the point Q(0, ) is on the line. If we set y to 0, we get for x. So the point R( ,0) is on the line. If we set x to 1, we get y = 1. So the point S(1,1) is on the line. If we set y to -1, we get x = 4. So the point T(4,-1) is on the line. If we set x to -2, we get y = 3. So the point U(-2,3) is on the line. (ii) The points we found in (i) are: Q(0, ), R( ,0) , S(1,1), T(4.-1), and U(-2,3). Using Q and R, the ratio: . Using Q and S: the ratio: . Using Q and T: the ratio: . Using Q and U: the ratio: . If we go on and calculate the ratios (do it!) for all pairs of points among Q, R, S, T, and U, we would get the same value . This confirms the fact that, the ratio is always the same for any two points on a line. That ratio is the slope of the line.
Note 2 If the equation of a line is given in standard form: , then its slope is , provided B is not 0. Indeed, suppose that B is other than 0. Setting x to 0 and solving for y, we get y = . That is, the point Q (0, ) is on the line. Setting x to 1 and solving for y, we get y = . So that the point R(1, ) is also on the line. Using the points Q and R, we can calculate the slope as the rise over the run: , as claimed.
Refer to the interactive exploration below.
Part 1. When you select two distinct points on the Cartsian Coordinate System, the line passing through those two points is drawn, its slope is calculated and displayed, an its equation in standard form is calculated and displayed. · Experiment with this functionality until you are comfortable with calculating the slope and the equation of a line in standard form given two of its points.
Part 2. There are three scroll bars labeled “A”, “B”, and “C”. The values displayed on those scroll bars define the equation of a line in standard form: . As the values of A, B, and C are changed continuously on the scroll bars, the graphs of the corresponding lines are displayed accordingly. · Change the value of A on its scroll bar (continuously) and observe the affect of changing its value has on the line. Write a sentence or two describing what you observe. What happens when A = 0? · Change the value of B on its scroll bar (continuously) and observe the affect of changing its value on the line. Write a sentence or two describing what you observe. What happens if B = 0? · Change the value of C on its scroll bar (continuously) and observe the affect of changing its value has on the line. Write a sentence or two describing what you observe.
You may also enter the value of A, B and C directly in the corresponding text fields directly below the scroll bars and then push the button “graph line” to draw the line. Use this functionality to graph the following lines: (i) (ii) (iii)
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