okay i really don't understand this problem.
It says:
Prove the point-slope equation property which is :
Point-slope equation(an nonvertical line with slope m and containing a point (X1,Y1) has the point-slope equation of y-y1=m(x-x1).
Now show that any point (x2,y2) on the line described must satisfy the equation, and that any point satisfying the equation must be on the line described.
To prove the point-slope equation property, we need to show two things:
1. Any point (x2, y2) on the line described must satisfy the equation y - y1 = m(x - x1).
2. Any point (x2, y2) that satisfies the equation y - y1 = m(x - x1) must be on the line described.
Let's start with the first part:
1. To show that any point (x2, y2) on the line described satisfies the equation y - y1 = m(x - x1), we can use the definition of slope. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1)/(x2 - x1)
Now, let's substitute the values of (x1, y1) into the equation:
m = (y2 - Y1)/(x2 - X1)
Cross-multiplying, we have:
(y2 - y1) = m(x2 - x1)
Rearranging the equation, we get:
y - y1 = m(x - x1)
This shows that any point (x2, y2) on the line described must satisfy the point-slope equation.
Now, let's move on to the second part:
2. To prove that any point (x2, y2) that satisfies the equation y - y1 = m(x - x1) must be on the line described, we will use a similar approach.
Starting with the equation y - y1 = m(x - x1), we will expand and rearrange it:
y - y1 = mx - mx1
Adding y1 to both sides:
y = mx - mx1 + y1
Rearranging the equation in the slope-intercept form, we get:
y = mx + (y1 - mx1)
Comparing this equation to the standard slope-intercept form y = mx + b, we can see that the y-intercept is (y1 - mx1).
Since any point on the line can be represented by (x, y), and we have shown that y = mx + (y1 - mx1), we can conclude that any point satisfying the equation must be on the line described.
Therefore, we have proven both parts of the point-slope equation property.