Graphing Linear Equations

Graphing Linear Equations exercise

• Describe how to solve linear equations by graphing.

  Hints

  The y-intercept doesn't depend on the variable $x$.

  The slope represents the change depending on the time.

  Depending on the time the drone can download a certain amount of data.

  Solution

  Dr. Evil wants to write the problem as a linear equation in slope-intercept form:

  $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

  What do we know?

  • We know the download rate of the drone is $1.25~PB$ per second. This is the slope $m$.
  • We also know the amount of storage space used for the hacking program is $15~PB$. This amount of space is independent of the time and is the y-intercept.

  The information written as a linear equation in slope-intercept form:

  $y=1.25x+15$.

  From the graph, you can see the drone C.H.E.E.S.E. 1 won't be able to download the entire amount data in the allotted amount of time.

• Does MAC2 have time enough to copy all the data?

  Hints

  The slope-intercept form is:

  $y=mx+b$.

  $m$ is the slope and $b$ is the y-intercept. In this case:

  $y=1.75x+15$.

  In order to draw the graph of a linear equation, you need to find the y-intercept first.

  In this case the slope is $m=1.75=\frac74$. This means to run four and rise seven.

  Determine the y-value for $x=60$.

  Solution

  Write an equation in slope-intercept form:

  $y=mx+b$

  For this situation, $m$ corresponds with the rate of copied data per second:

  $m=1.75=\frac74$

  $b$ corresponds with the required storage independent of time:

  $b=15$

  $y=1.75\times 60+15=120$.

  First we mark a point at the y-intercept ($b=15$) then move four to the right and rise seven - mark this point then draw a line to connect the two points.

  As you can see, the graph reflects the information for this situation written in slope-intercept form.

  At 60 seconds, the y-value is more than $105~PB$ ($90~PB$ of copied data + $15~PB$ of internal storage used for the hacking program).

  Therefore MAC 2 can successfully copy the full amount of data.

• Examine if C.H.E.E.S.E. 1 succeeds at getting $8~oz$ of cheese for Dr. Evil.

  Hints

  To calculate $b$, plug $x=0$ into $y=mx+b$.

  Keep in mind that the amount of cheese needed from the drone has a negative impact.

  The drone succeeds in getting more than 8 oz of cheese.

  Solution

  For this problem, Dr. Evil has to set up a linear equation written in slope-intercept form: $y=mx+b$.

  • $m$ is the slope
  • $b$ is the y-intercept

  Let's start with the y-intercept. This value doesn't depend on time and therefore it is a constant. This refers to the amount of cheese that the drone needs for itself:

  $b=-0.4$

  Now we still have to determine the slope - the rate of cheese per minute:

  $m=0.3$

  The line written in slope-intercept form:

  $y=0.3x-0.4$

  In order to find out whether C.H.E.E.S.E. 1 will succeed in getting 8 oz we still have to plug in the time $x=30$:

  $0.3\times30-0.4=9-0.4=8.6$.

  Great: the brunch is saved.

• Determine the linear equation.

  Hints

  You will find the y-intercept at the point where the line crosses the y-axis.

  The slope is equal to the change in $y$ divided by the change in $x$.

  Solution

  The y-intercept is quite easy to find. It is the point where the line crosses the y-axis:

  $b=-3$

  The slope is equal to the rise over the run.

  • rise: two steps up
  • run: one step to the right

  This result is $m=\frac21=2$.

  The information written as a linear equation in slope-intercept form:

  $y=2x-3$

  Let's check this equation. We can see on the right side, that the x-intercept is given by $x=1.5$.

  Now we plug in this value into the equation: $y=2\times 1.5-3=3-3=0$ $\surd$.

• Identify the different parts of the graph and the equation.

  Hints

  Remember the slope intercept form for linear equations:

  $y=mx+b$

  If you substitute $x$ with zero we get:

  $y=m\cdot 0+b=b$

  This the point, where the line intercepts the y-axis, the y-intercept.

  You can also remember the slope as rise over run.

  Solution

  Remember how to write a linear equation in slope-intercept form:

  $y=mx+b$.

  • The factor $m$ is the slope.
  • The term $b$ is the y-intercept.

  Let's analyze the linear equation $y=1.25x+15$:

  • The slope is $m=1.25$.
  • The y-intercept is $b=15$.

  The slope is the change of $y$ divided by the change of $x$. You can also think of this as rise over run. In this case, there is a rise of $5$ (change in $y$) and a run of $4$ (change in $x$).

  To draw the slope it`s better to change a decimal into a fraction:

  $1.25=\large{\frac{125}{100}}=\large{\frac{5}{4}}$

• Decide which equation belongs to which graph.

  Hints

  Each equation is written as a linear equation in slope-intercept form:

  $y=mx+b$.

  The y-intercept is easy to find. It is the point where the line crosses the y-axis.

  The slope is equal to the rise over the run.

  • The rise goes up or down.
  • The run goes to the right or to the left.

  Solution

  Each equation is written in slope-intercept form $y=mx+b$.

  The blue line as well as the green line have the same y-intercept:

  $b=3$

  The yellow line as well as the red line have the same y-intercept:

  $b=-1$

  Now we determine the slope for each line:

  The red line rises one step and runs four steps to the right. This results in $m=\frac14=0.25$. We can write the equation:

  $y=0.25x-1$

  $~$

  The yellow line rises one step and runs one step to the right. This results in $m=\frac11=1$. We can write the equation:

  $y=x-1$

  $~$

  The green line rises one step and runs four steps to the right. This results in $m=\frac14=0.25$. We can write the equation:

  $y=0.25x+3$ This line has the same slope as the red one. These lines are parallel.

  $~$

  The blue line rises one step and runs two steps to the right. This results in $m=\frac12=0.5$. We can write the equation:

  $y=0.5x+3$