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*Two equivalent point slope form equations for the line passing through (-4,2) and (-2,16)

**y-16 = 7 (x+2)**

**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
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**infinite equivalent point slope form equations***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!