Dealing with slope.
Solution: y-2 = 7 (x+4) or y-16 = 7 (x+2) To understand why, please read the following step by step solution.
STEP 1: Read, understand the situation within, identify and pull out important information.
• The point slope form equation: y-y1 = m (x-x1); m=slope; (x1,y1) are the coordinates of a point of the line.
• There is only one line passing through two points.
• A line is a set of infinite points. So infinite pairs of points can determine the same line with the slope “m”.
• For the point slope form equation we need one point of the line (we are giving two of them!) and the slope m.
• Determine the slope “m” using the two given points of the line.
STEP 2: Translate keywords to their mathematical symbols:
• Slope of the line: m = (y2-y1) / (x2-x1) , where (x1, y1) and (x2, y2) are the coordinate of the two giving points.
STEP 3: Set up and solve the equation or problem:
• First determine the slope:
m= (16-2) / (-2-(-4)) or m = (2-16) / (-4-(-2) Being the points on the same line, it does not matter the order you select them to determine the slope!
m = 14 / (-2+4) or m = (-14) / (-4+2)
m = 14/ 2 or m = (-14) / (-2)
m = 7 or m = 7
• Write the point slope form equation: y-y1=m(x-x1) or y-y2 =m(x-x2)
y-2=7(x-(-4)) or y-16=7(x-(-2)) by selecting (-4,2) or (-2,16)
y-2 = 7 (x+4) or y-16 = 7 (x+2) Two equivalent point slope form equations for the line passing through (-4,2) and (-2,16)
STEP 4: Check the solution:
y-2=7(x+4) or y-16=7 (x+2) Substitute the coordinates of one given point (-4,2)
2-2 = 7 (-4+4) or 2-16 = 7 (-4+2) Applying the Distributive Property
0 = -28 + 28 or -14 = -28 + 14
0 = 0 or -14 = -14 Both are identities, so the two equations represent the point slope form equation of the line through the points (-4,2), (-2,16).
STEP 5: Curiosities
- There are infinite equivalent point slope form equations representing the same line! In fact, meanwhile “m” remains always the same (it’s the same line!), by selecting a different point of the line, we’re just changing the coordinates (x1,y1), and so we are getting another equation. Even though those equations look different (because we are considering different coordinates), they are equivalent!
- The intercept slope form helps us to better understand it. Let’s write both point slope form equations in intercept slope form:
y-2=7(x+4); y-2=7x+28; y=7x+30 It’s the Intercept Slope form
y-16=7(x+2); y-16=7x+14; y=7x+30 It's the same Intercept Slope form!
The intercept slope form equation is just unique in representing a line because it has only one y-intercept point!