Write Equations Using \(y=mx+b\)
Consider the graph below containing the points \((-4, -2)\) and \((-2, 3)\).
- What is the slope of the line containing these two points?
- If we use the slope to count the rise and run, what are the coordinates of the \(y\)-intercept?
- Write the equation of this line in slope-intercept form.
Remember: We can determine the slope between any two points using the slope-formula. Given Point A: \((-4, -2)\) and Point B: \((-2, 3)\) we calculate the slope as:
\[slope=\dfrac{3-(-2)}{-2-(-4)}=\dfrac{3+2}{-2+4}=\dfrac{5}{2}\]
Additionally, we can use the slope-intercept form of the line to determine the \(y\)-intercept, as shown below. Finding the \(y\)-intercept in this manner is useful when counting using the slope is impractical or cumbersome.
By plugging in the slope for \(m\), and one point on the line for \(x\) and \(y\), we can find the \(y\)-intercept.
\[\begin{align} y&=mx+b \\\\ 2&=\dfrac{5}{2}(-4)+b \\\\ 2&=-10+b \\\\ 8&=b\end{align}\]
Therefore the equation of the line is: \(y=\dfrac{5}{2}x+8\).
- Use this method to find the equation of the line between the points \(( -2, -1)\) and \((3, 2)\).
