Computing the Slope of a Line

Computing the Slope of a Line exercise

• Describe how to find the slope of the line given two points.

  Hints

  Keep rise over run in mind to remember the order of division in the slope formula.

  Pay attention to the order of subtraction.

  Solution

  The slope of any non vertical line can be found using the slope formula.

  Remember: rise over run.

  The slope formula is given by dividing the change in $y$ by the change in $x$:

  $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$.

  We can now apply this formula to the two given points $A(0,0)$ and $B(2,8)$:

  $m=\frac{8-0}{2-0}=\frac82=4$.

• Find the slope of the line given a graph.

  Hints

  The slope tree helps you to remember the slope of different lines.

  Use the slope formula: $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

  An increasing line has a positive slope and a decreasing line has a negative slope.

  Solution

  The slope formula is given by

  $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$.

  With this we can determine the slopes of the given lines:

  The red line:

  You can read off the points $(1,2)$ and $(4,5)$:

  $m=\frac{5-2}{4-1}=\frac{3}{3}=1$.

  The blue line:

  Here you can read off the points $(0,6)$ and $(1,3)$:

  $m=\frac{3-6}{1-0}=\frac{-3}{1}=-3$.

  The green line:

  First, read off the points $(1,3)$ and $(4,3)$:

  $m=\frac{3-3}{4-1}=\frac{0}{3}=0$.

  The orange line:

  The readable points are $(2,4)$ and $(4,3)$:

  $m=\frac{3-4}{4-2}=\frac{-1}{2}=-0.5$.

• Determine the slope of the lines given two points.

  Hints

  Use the slope formula: $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

  Keep the order of subtraction in mind when using the slope formula.

  Pay attention when subtracting negative numbers. Look at the example beside with the points $(-1,-1)$ and $(3,1)$.

  Solution

  The slope formula is given by

  $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

  With this, we can determine the slopes of the given lines:

  • $\mathbf{(3,1)}$ and $\mathbf{(5,7)}$ give a slope of $m=\frac{7-1}{5-2}=\frac{6}{3}=2$

  • $\mathbf{(-1,7)}$ and $\mathbf{(3,5)}$ give a slope of $m=\frac{5-7}{3-(-1)}=\frac{-2}{4}=-0.5$

  • $\mathbf{(4,3)}$ and $\mathbf{(3,4)}$ give a slope of $m=\frac{4-3}{3-4}=\frac{1}{-1}=-1$

  • $\mathbf{(-1,-1)}$ and $\mathbf{(2,5)}$ give a slope of $m=\frac{5-(-1)}{2-(-1)}=\frac{6}{3}=2$

  • $\mathbf{(2,2)}$ and $\mathbf{(4,9)}$ give a slope of $m=\frac{9-2}{4-2}=\frac{7}{2}=3.5$

• Calculate the slope of the following situations.

  Hints

  For any two given points, use the slope formula:

  $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

  Pay attention to the order of subtraction when using the slope formula.

  Solution

  First, identify which value is independent (i.e. the $x$-axis) and which value is dependent (i.e. the $y$-axis).

  Then, identify two points from the given information which we can use to plug into the slope formula,

  $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

  • Climber: $m=\frac{750-250}{7-2}=\frac{500}{5}=100$ feet per hour

  • Mr. Miller: $m=\frac{450-0}{6-0}=\frac{450}{6}=75$ miles per hour

  • Paul: $m=\frac{140-160}{10-0}=\frac{-20}{10}=-2$ lbs per week

  • Luke: $m=\frac{27-25}{5-1}=\frac24=0.5$ dollars per day

  In all the given word problems, you can interpret the slope as the average change or rate depending on time.

• Identify some points on the line.

  Hints

  The first coordinate of a point is the $x$-coordinate and the second is the $y$-coordinate.

  Draw a line from the point parallel to the $x$-axis and read off the $y$-coordinate: this is where the line passes the $y$-axis.

  Similarly, you get the $x$-coordinate.

  Solution

  To get the coordinates of any point you proceed as follows:

  • Draw a line from the point parallel to the x-axis and read off the y-coordinate: This is where the line passes the y-axis.
  • Draw a line from the point parallel to the y-axis and read off the x-coordinate: This is where the line passes the x-axis.

  So you get, from left to the right, the following points:

  • $(0,1)$
  • $(2,2)$
  • $(4,3)$
  • $(8,5)$

• Identify the equations for the given graphs in slope intercept form.

  Hints

  First, you have to read off two points lying on the line.

  Remember, the general form of a linear equation is $y=mx+b$.

  You get the slope $m$ by using the slope formula,

  $\quad~~~m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$.

  The $y$-intercept $b$ is the point where the line intersects the $y$-axis.

  • Increasing lines have a positive slope.
  • Decreasing lines have a negative slope.

  Solution

  The general form of a linear equation is $y=mx+b$.

  • $b$ is the $y$-intercept, which is the point where the line intersects the $y$-axis.
  • $m$ is the slope.

  To determine the slope, we use the slope formula,

  $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$.

  Let's start with the green line and the points $(0,3)$ and $(2,7)$. We know from the point $(0,3)$ that $b=3$. Now we determine the slope:

  $m=\frac{7-3}{2-0}=\frac{4}{2}=2$.

  Thus we have the following equation: $y=2x+3$.

  Now we take a look at the blue line and the points $(0,3)$ and $(2,0)$. Again we can read off $b=3$ directly. We still have to compute the slope:

  $m=\frac{0-3}{2-0}=\frac{-3}{2}=-1.5$.

  Together we get the equation: $y=-1.5x+3$.

  The next one is the red line and the points $(0,2)$ and $(4,4)$. The first point with $x=0$ gives us $b=2$. Almost done: just the slope is left:

  $m=\frac{4-2}{4-0}=\frac{2}{4}=\frac12$.

  So we have that the equation is $y=0.5x+2$.

  Last but not least, we have to handle the violet line and the points $(0,2)$ and $(4,-1)$. We can deduce from the point $(0,2)$ that $b=2$. So the slope is,

  $m=\frac{-1-2}{4-0}=\frac{-3}{4}=-0.75$,

  and the equation is given by: $y=-0.75x+2$.