Graphing Linear Equations 04:31 minutes

Video Transcript

Transcript Graphing Linear Equations

Dr. Evil is in his elegant lair busily hatching another especially evil plot. This time, he’s after some information stored on General Good’s computer. To steal the data, he plans to use C.H.E.E.S.E.1, a computer hacking drone.

Before Dr. Evil can launch his plan, there are some problems he must solve. C.H.E.E.S.E.1 can copy data at a speed of one point two-five petabytes per second.

Dr. Evil has been surveilling General Good and knows that every day, exactly at midnight, General Good opens the window and goes into the hallway to stretch for 60 seconds. So the drone will have just sixty seconds to download all of the data. The problem is, can C.H.E.E.S.E.1 complete the task in time?

Writing the Linear Equation in Slope-Intercept Form

You may ask, how can Dr. Evil solve this problem? He can graph linear equations using slope-intercept form. Let’s take a look at the coordinate system.

The file he plans to steal is ninety petabytes. The drone has exactly sixty seconds to copy all of the data and the hacking program loaded on C.H.E.E.S.E.1 takes up 15 petabytes of space. The drone will have a total of 105 petabytes of data after copying the wanted files, if the caper is successful.

Now, he writes the equation in slope-intercept form: y = mx + b. Remember, m = slope and b = y-intercept. For this equation, x represents the number of seconds it takes for the drone to download the data and y represents the total amount of data stored on the drone.

The drone can download 1.25 petabytes of data per second, so the coefficient – or the slope – of x is 1.25.

The hacking program loaded on C.H.E.E.S.E.1 takes up 15 petabytes of space. Since this number does not change, it's called a constant. It is also the y-intercept.

Putting all of this together, the equation in slope-intercept form is y = 1.25x + 15.

Graphing the Linear Equation

Dr. Evil graphs the equation. The line touches the y-axis at the y-intercept, 15. The slope is 1.25. To graph this, write this decimal as a fraction. So the slope is 5/4, which means that the line rises 5 and runs 4. Dr.

Evil needs to know if C.H.E.E.S.E.1 do the task. Oh no! The line doesn’t go through the ordered pair of (60, 105) representing sixty seconds and 105 petabytes, respectively.

C.H.E.E.S.E.1 is too slow. What can Dr. Evil do? Dr. Evil has a plan B. He can use MAC2, a faster version. It’s a bit of a risk cause MAC2 is a beta version, but Dr. Evil is desperate. MAC2 has a download speed of one point 75 petabytes per second.

Using the download speed for MAC2, Dr. Evil writes a new equation in slope-intercept form. This time, the x coefficient, or slope, is equal to 1.75. The y-intercept does not change.

Dr. Evil gathers his evil data and draws an evil graph. The line still touches the y-axis at 15, but since the slope is 1.75 or 7 over 4 in fraction form, the line rises 7 and runs 4.

The graph indicates MAC2 can download 120 petabytes of data in sixty seconds. Dr. Evil is ecstatic! The graph shows MAC2 can do the task with seconds to spare!

Dr. Evil can’t wait to take a look at the data stolen from General Good. Oh no! What’s this? MAC2 copied the wrong file. Now, Dr. Evil’s computer is infected with the Kitty Cat virus. Meow!

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Graphing Linear Equations Übung

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  • Describe how to solve linear equations by graphing.

    Tipps

    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.

    Lösung

    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.

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

    Tipps

    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.

    Lösung

    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:

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

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

    Tipps

    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 $x=60$.

    Lösung

    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.

  • Decide which equation belongs to which graph.

    Tipps

    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.

    Lösung

    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:

    $y=0.5x+3$

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

    Tipps

    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.

    Lösung

    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.

    Tipps

    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 $x$.

    Lösung

    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: $y=2\times 1.5-3=3-3=0$ $\surd$.