The Slope of the Line y=mx+b 07:37 minutes

Video Transcript

Transcript The Slope of the Line y=mx+b

There's trouble on the set of FISTS OF DANGER, the new, big-budget action movie from Furious Films. The movie is missing something...but what? According to the latest research, adding scenes with adorable animals to any movie will increase ticket sales! This may sound like a weird idea for an action film but with some cute animals and the slope of the line y = mx + b, we could save this movie yet! Studies show that for each second of baby koala footage featured in a film, ticket sales go up.

This linear relationship is expressed in the following equation: 3y equals 500x where 'x' represents seconds of koala footage and 'y' is the number of tickets sold. To better understand this equation, let’s rewrite it using slope-intercept form, or 'y equals mx + b'. Using this form, it will be easier for us to understand how 'x' and 'y' change in relation to each other. To get our equation in this form, all we have to do is isolate the variable, 'y'. To get 'y' by itself, we can divide both sides of the equation by 3, thanks to the division property of equality. Great! Now our equation is in slope-intercept form, so we can easily identify the slope, 'm', which will always be the coefficient of 'x'. Here, the slope 'm' equals 500 over 3, or 500 thirds. Remember that slope represents the change in 'y' over the change in 'x'.So that means, for every 3 seconds of footage of baby koalas, we sell 500 more tickets.

Wait, what about the 'b' in our 'y equals mx + b'? The 'b' term tells us the y-intercept, or the value of 'y' when 'x' equals zero. It may not look like it, but our equation actually DOES have a 'b' term. Here, 'b' is just zero. So that means when 'x' is 0, 'y' is also 0, so the graph of this line passes through the origin at (0, 0). To sketch this graph, start with a known point on the line, like (0, 0). Then go up and over according to your slope. Up for change in 'y', over for change in 'x'. Since our slope is 500 over 3, we go up 500 and over 3. Now draw a straight line through the points! Whoa, we've got to get some more baby koalas in on this project!

What other animals can we get on board? Research also shows that having a miniature teacup pig on screen can boost ticket sales. This can be represented by using the equation, 120x minus 4y equals 20 where 'x' represents seconds of pig footage and 'y' is the number of tickets sold. Let's transform this equation into 'y equals mx + b' so we can figure out how the teacup pigs are going to affect ticket sales. To get 'y' alone, subtract 120x from both sides and divide every term in the equation by negative 4, to cancel out the coefficient of 'y'. Finally, we can rearrange the terms on the right side of the equation so it matches the more familiar form of 'y equals mx plus b' that we recognize. Now we can easily identify the slope, 'm', as the coefficient of 'x'. The slope tells us that for every second of teacup pig footage, we sell 30 more tickets. We can also easily identify 'b', or the y-intercept, at (0, -5). To sketch this graph, let's start with the point we just found. Now use what we know about the slope and go up 30, to the right 1, and draw a line through these points. Those pigs have got a future in Hollywood!

But not all animals are so good for the movies. The notorious honey badger has a crippling effect on ticket sales. Let’s take a look at this formula. 3x plus 5y equals 10 where 'x' represents seconds of honey badger footage and 'y' is the number of tickets sold. Let’s isolate the variable, 'y'. Start by subtracting 3x from both sides and divide all the terms in the equation by 5. Much better! The slope in this equation is negative 3 over 5. This means for every 5 seconds of honey badger screen time, the movie actually sells 3 fewer tickets. We can also easily identify the y-intercept, or 'b', at (0, 2). So how do we graph this line? Start with a point you know, like the y-intercept at (0, 2). Because the slope is negative, we then go down 3, to the right 5. Now we draw a line through these points man those badgers are really dragging sales down!

To review: We can write linear equations using the form 'y = mx + b'. This is known as Slope-Intercept Form. In this form, 'm' represents the slope the line, or change in 'y' over change in 'x' and 'b' is the y-intercept, or where the line crosses the y-axis.

Using the market research, the studio executives have gone ahead and reshot their film using a new title... …TROUBLE CUTIES?´Man, Hollywood really has changed these cute little guys...

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  • Recall how to write equations in slope-intercept form.

    Tipps

    Isolate the variable $y$ to put the linear relationship into slope-intercept form.

    Subtract the $x$ term from both sides of the equation.

    Divide every term by the coefficient of $y$ to cancel it out.

    Lösung

    All linear relationships can be written in slope-intercept form, $y=mx+b$, by isolating the variable $y$.

    1. For the koala example, the linear relationship is $3y=500x$.

    • Divide both sides by $3$.
    • $y=\frac{500}{3}x$
    2. For the pig example, the linear relationship is $120x-4y=20$.
    • Subtract $120x$ from both sides of the equation.
    • $-4y=20-120x$
    • Divide all terms by $-4$.
    • $y=-5+30x$.
    • Rearrange the terms on the right.
    • $y=30x-5$
    3. For the honey badger example, the linear relationship is $3x+5y=10$.
    • Subtract $3x$ from both sides of the equation.
    • $5y=10-3x$.
    • Divide all terms by 5.
    • $y=2-\frac{3}{5}x$
    • Rearrange the terms on the right.
    • $y=-\frac{3}{5}x+2$

  • Determine if the equation is written in slope-intercept form or not.

    Tipps

    Slope-intercept form is $y=mx+b$.

    Slope-intercept form can also look like $y=b+mx$, $mx+b=y$ or $b+mx=y$.

    If there is no $y$-intercept $b$, slope-intercept form will look like $y=mx$.

    Lösung

    Slope-intercept form is $y=mx+b$. However, the terms can be rearranged and still be in slope-intercept form. The following equations are $4$ different ways to represent slope-intercept form.

    • $y=mx+b$: the original slope-intercept form
    • $y=b+mx$: the terms on the right side are switched
    • $mx+b=y$: the equation is flipped so that y is on the right
    • $b+mx=y$: the equation is flipped with y on the right, and the $mx$ and $b$ terms are switched
    If there is no $y$-intercept, the equation will look like $y=mx$ or $mx=y$.

    The following equations are in slope-intercept form because they match one of the equations listed above.

    • $y=5x-3$
    • $y=4+2x$
    • $y=-\frac{2}{5}x$
    • $y=-4x$
    • $1-3x=y$
    • $-x+1=y$
    The following equations are not in slope-intercept form because they do not match any of the slope-intercept form equations above.
    • $x=-\frac{1}{2}y+1$
    • $\frac{3}{2}x+y=2$
    • $2y=x+1$
    • $x=-\frac{1}{2}y+1$
    • $-y=2x-3$
    • $7y=2x$

  • Identify the slope and the $y$-intercept of each equation.

    Tipps

    Slope-intercept form is $y=mx+b$.

    The slope is $m$ and the $y$-intercept is $b$.

    To get $y$ alone, subtract the $x$ term if necessary, then divide each term by the coefficient of $y$.

    Lösung

    Slope-intercept form is $y=mx+b$ where $m$ is the slope and $b$ is the $y$-intercept. To find $m$ and $b$, put all of the equations in slope-intercept form $y=mx+b$.

    • $y=2x+1$ is in slope-intercept form. Therefore, the slope is $2$ and the $y$-intercept is $1$.
    • $y=\frac{8}{5}x$ is in slope-intercept form. Therefore, the slope is $\frac{8}{5}$ and the $y$-intercept is $0$ since there is no $b$ in the equation.
    • $y=-\frac{1}{2}x-3$ is in slope-intercept form. Therefore the slope is $-\frac{1}{2}$ and the $y$-intercept is -3.
    • $4x+y=1$ is not in slope-intercept form. Subtracting $4x$ from both sides of the equation yields $y=1-4x$. Then, rearrange the terms on the right so the equation is $y=-4x+1$. Now it looks like $y=mx+b$ which means the slope is $-4$ and the $y$-intercept is $1$.
    • $-2y=3x-6$ is not in slope-intercept form. To get $y$ alone, divide every term by $-2$: $y=-\frac{3}{2}+3$. The equation is now in the form $y=mx+b$ which means the slope is $-\frac{3}{2}$ and the $y$-intercept is $3$.
  • Determine the line.

    Tipps

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

    If $x=0$ in the point $(x,y)$, then the $y$-coordinate is the $y$-intercept $b$.

    If $x\neq 0$ in the point $(x,y)$, substitute the slope $m$ and the point $(x,y)$ into $y=mx+b$, and solve for $b$.

    Lösung

    Slope-intercept form, $y=mx+b$, consists of a point $(x,y)$, a slope $m$, and a $y$-intercept $b$. Given a point and a slope you can uniquely determine a line.

    1. $(0,-4)$, $m=\frac{1}{2}$

    • Since $x=0$, we know that the $y$-intercept is $-4$, $b=-4$
    • Now we can substitute $m$ and $b$ into $y=mx+b$
    • $y=\frac{1}{2}x-4$
    2. $(0,\frac{35}{2})$, $m=\frac{3}{5}$
    • Since $x=0$, we know that the $y$-intercept is $\frac{35}{2}$, $b=\frac{35}{2}$
    • Now we can substitute $m$ and $b$ into $y=mx+b$
    • $y=\frac{3}{5}x+\frac{35}{2}$
    3. $(1,2)$, $m=2$
    • Since $x\neq 0$, we substitute the values of $x$, $y$, and $m$ into $y=mx+b$, to solve for $b$.
    • $2=2(1)+b$
    • $2=2+b$
    • $0=b$
    • Now that we know $b=0$, we can substitute $b$ and $m$ into $y=mx+b$
    • $y=2x+0$
    • $y=2x$
    4. $(-1,-3)$, $m=-2$
    • Since $x\neq 0$, we substitute the values of $x$, $y$, and $m$ into $y=mx+b$, to solve for $b$.
    • $-3=-2(-1)+b$
    • $-3=2+b$
    • $-5=b$
    • Now that we know $b=-5$, we can substitute $b$ and $m$ into $y=mx+b$
    • $y=-2x-5$
    5. $(0,-6)$, $m=-\frac{17}{3}$
    • Since $x=0$, we know that the $y$-intercept is $-6$, $b=-6$
    • Now we can substitute $m$ and $b$ into $y=mx+b$
    • $y=-\frac{17}{3}x-6$
    6. $(4,4)$, $m=6$
    • Since $x\neq 0$, we substitute the values of $x$, $y$, and $m$ into $y=mx+b$, to solve for $b$.
    • $4=6(4)+b$
    • $4=24+b$
    • $-20=b$
    • Now that we know $b=-20$, we can substitute $b$ and $m$ into $y=mx+b$
    • $y=6x-20$
    7. $(-4,3)$, $m=\frac{1}{4}$
    • Since $x\neq 0$, we substitute the values of $x$, $y$, and $m$ into $y=mx+b$, to solve for $b$.
    • $3=\frac{1}{4}(-4)+b$
    • $3= -1+b$
    • $4=b$
    • Now that we know $b=4$, we can substitute $b$ and $m$ into $y=mx+b$
    • $y=\frac{1}{4}x+4$

  • Match the equations with their slope-intercept form.

    Tipps

    Isolate the variable $y$ by subtracting the $x$ term from both sides.

    Rearrange the terms or flip the equation to make it look like $y=mx+b$.

    Divide every term by the coefficient of $y$.

    Lösung

    1. $-2y+3x=8$

    • Subtract $3x$ from both sides
    • $-2y=8-3x$
    • Divide every term by $-2$
    • $y=-4+\frac{3}{2}x$
    • Rearrange the terms to look like $y=mx+b$
    • $y=\frac{3}{2}x-4$
    2. $12x+3y=6$
    • Subtract $12x$ from both sides
    • $3y=6-12x$
    • Divide every term by $3$
    • $y=2-4x$
    • Rearrange the terms on the right
    • $y=-4x+2$
    3. $\frac{1}{3}y=2x$
    • Multiply both sides by $3$
    • $y=6x$
    4. $5y=20x+30$
    • Divide every term by 5
    • $y=4x+6$
    5. $9x+8=3y$
    • Flip the equation
    • $3y=9x+8$
    • Divide every term by $3$
    • $y=3x+\frac{8}{3}$
    6. $4x=6y$
    • Flip the equation
    • $6y=4x$
    • Divide both sides by $6$ and simplify
    • $y=\frac{2}{3}$

  • Graph the equation $-y-3x+4=0$.

    Tipps

    The graph of a line is $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.

    To graph the $y$-intercept, plug $x=0$ into the equation.

    The slope, $m$, is the change in $y$ divided by the change in $x$.

    Lösung

    The line $-y-3x+4=0$ can be rewritten in slope-intercept form by solving for $y$.

    $~$

    To solve for $y$:

    1. Add $3x$ to both sides: $-y+4=3x$.

    2. Subtract $4$ from both sides: $-y=3x-4$.

    3. Divide both sides by $-1$: $y=-3x+4$.

    $~$

    To identify the slope and $y$-intercept:

    • The slope is $m=-3$.
    • The $y$-intercept is $b=4$.
    $~$

    To identify the point at $x=1$:

    • Start at the $y$-intercept, $(0,4)$.
    • As the slope is $m=-3$, if we subtract $3$ from the $y$-coordinate and add $1$ to the $x$-coordinate of a point on our line, we get another point on our line.
    • Doing this with our $y$-intercept, we get $(0+1,4-3)=(1,1)$, which is the point on our line at $x=1$.
    • Use the slope $m=-3$ to find the point at $x=1$.
    $~$

    To graph:

    1. Plot the $y$-intercept $(0,4)$.

    2. Plot the point $(1,1)$.

    3. Draw a line connecting these two points.