Solution: y-5=3(x-2) or y-2=3(x-1) in point slope form; y=3x-1 in intercept slope form
To understand why, please read the following step by step solution.
STEP 1: Read, understand the situation within, identify and pull out important information.
• Consider the point slope form and the intercept slope form equations.
• Both equations need the slope “m”, and the coordinates of a point on the line.
• Calculate the slope “m” using the coordinates of the two given points (2,5) and (1,2).
• Slope “m”= RISE/RUN; be congruent when going from one point to another point of the line. You have to get the same result.
RISE= Change in y: (y2-y1); RUN= Change in x: (x2-x1). You’re going from P1 to P2 (ending point)
RISE= Change in y: (y1-y2); RUN= Change in x: (x1-x2). You’re going from P2 to P1 (ending point)
• For the point slope form: use the coordinates of one of the two given points. If you chose the other point (or any other point of the line) you’ll get an equivalent point slope equation for the same line!
• For the intercept slope form: we would need the y-intercept, but we don't have the point where the line crosses the y-axis. We can get the intercept slope form from the point slope form by isolating "y". Remember: the slope intercept form is a particular case of the point slope form, because the point used is the y-intercept point (0,b).
STEP 2: Translate keywords to their mathematical symbols:
• Point slope form equation: y-y1=m(x-x1) or y-y2=m(x-x2) It does not matter!
• Intercept slope form equation: y=mx+b b is the y-intercept of the intercept point (0,y)
• Slope of the line: m = (y2-y1) / (x2-x1) or m = (y1-y2) / (x1-x2) It doesn't matter, just be congruent!
STEP 3: Set up and solve the equation or problem:
• Slope “m”:
m= (2-5) / (1-2) or m = (5-2) / (2-1)
m = -3 / (-1) or m = 3 / 1
m = 3 or m = 3
• Point slope form equation:
y-y1 = m (x-x1) or y-y2 = m (x-x2)
y-5 = 3 (x-2) using (2,5) or y-2 = 3 (x-1) using (1,2)
These are two point slope form equations for the line passing through (2,5) and (1,2)
• Intercept slope form equation: (Starting from the point slope form equation)
y-5 = 3 (x-2) or y-2 = 3 (x-1) apply the Distributive Property of Multiplication
y-5 = 3x-6 or y-2 = 3x-3
+5 +5 or +2 +2 to isolate "y"
y = 3x-1 or y = 3x-1 This is the intercept slope form equation for the line through (2,5) and (1,2)
STEP 4: Check the solution:
• For the point slope form equation:
y-5 = 3 (x-2) or y-2 = 3 (x-1)
5-5 = 3 (2-2) or 5-2 = 3 (2-1) substituting the coordinates of the given point (2,5)
0 = 3 (0) or 3 = 3 (1) Applying the Distributive Property of Multiplication
0 = 0 or 3 = 3 Both are identities, so the two equations represent the point slope form equations of the same line through the points (2,5), (1,2).
• For the intercept slope form equation:
y = 3x-1 or y = 3x-1
5 = 3(2)-1 or 2 = 3(1)-1 substituting the coordinates (2,5) and (1,2) respectively
5 = 6-1 or 2 = 3-1
5 = 5 or 2 = 2 Both are identities, so the equation is the intercept slope form equation of the line through the points (2,5), (1,2).
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 are just changing the coordinates (x1,y1), and so we are getting another equation. Even though these equations look different they are equivalent.
- The intercept slope form equation is just unique in representing a line because it has only one y-intercept point !
- Do you want to graph this line? Just plot the two given points (2,5) and (1,2) in a (x,y) Cartesian plane. Then draw a line through these two points! REMEMBER: there is only one line passing through two points. Two points define a unique line!
- Getting the slope intercept form from the point slope form:
y-y1 = m(x-x1) Substitute the y-intercept point coordinates (0.b)
y-b = m(x-0)
y = mx - m(0) + b
y = mx + b tutto bene!