See in the explanation
If you have two variables, the proportional relationship tells us that the ratio between these variables stays the same. Suppose you have two variable [tex]x \ and \ y[/tex], so in proportional relationship we say [tex]y[/tex] varies proportionally as [tex]x[/tex] or in other terms [tex]y[/tex] is directly proportional to [tex]x[/tex]. If so, then [tex]y[/tex] increases as [tex]x[/tex] increases or [tex]y[/tex] decrases as [tex]x[/tex] decreases:
[tex]y=kx \\ \\ k:Constant \ of \ proportionality[/tex]
But how does the slope relate to the equation? Well, in this case [tex]y=kx[/tex] is the equation of a line and [tex]k[/tex] which is the constant of proportionality represents the slope of the line.
Here we have:
[tex]y=mx \\ \\ Where: \\ \\ m:Slope[/tex]
The slope is the steepness of the line and also tells us its direction. But what do the graphs of lines given by that form have in common? Well, the common thing is that they all pass through the origin (0, 0), because the y-intercept is [tex]b=0[/tex]
On the other hand, how might they differ? Well, they might differ in the inclination. So we can measure this as:
The table shows relationship between time (s) and distance (m) of a train that is traveling. We know that the table represents a linear relationship because the between two consecutive outputs (distance) is constant, that is:
[tex]50-25=25 \\ 75-50=25 \\ 100-75=25[/tex]
[tex]y-y_{1}=m(x-x_{1}) \\ \\ m:Slope \\ \\ (x_{1},y_{1}):A \ point \ on \ the \ line \\ \\ \\ So: \\ \\ m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \\ Two \ points: \\ \\ (x_{1},y_{1})=(2,25) \\ (x_{2},y_{2})=(4,50) \\ \\ m=\frac{50-25}{4-2} \\ \\ m=\frac{25}{2}[/tex]
So, the equation of the line is:
[tex]\boxed{y-25=\frac{25}{2}(x-2)}[/tex]
a)
In this case, we have a relationship between a hiker's elevation and time given by a proportional relationship. We know this is so because the graph is a line and passes through the origin (See Part 2). So we can find the constant of proportionality by computing the slope of the line. Therefore:
[tex]k=m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \\ Two \ points \ are: \\ \\ (x_{1},y_{1})=(0,0) \\ (x_{2},y_{2})=(1,30) \\ \\ k=\frac{30-0}{1-0} \\ \\ \boxed{k=30}[/tex]
So the slope is also:
[tex]\boxed{m=30}[/tex]
b)
So the equation of the line can be written as:
[tex]y=kx \\ \\ \boxed{y=30x}[/tex]
As we said before, the graph of lines given by the form:
[tex]y=mx[/tex]
passes through the origin having slope [tex]m[/tex]. In this case, we have the following equation:
[tex]y=-\frac{1}{2}x[/tex]
So we can graph this with two points, since the origin is a point that lies on the line, we just need another point. Let's say:
[tex]x=1 \\ \\ y=-\frac{1}{2}x \\ \\ y=-\frac{1}{2}(1) \\ \\ y=-\frac{1}{2} \\ \\ Another \ point \ is: \\ \\ (1,-1/2)[/tex]
Graph a line that passes through these two points as shown below
As the problem states, Resting heart rate is a measure of how fast the heart beats when a person is not performing a physical activity. So you can see that the graph shows the number of heartbeats over time for a given person. So we have the following equation of the slope in an incomplete form:
[tex]m=\frac{280-()}{()-2} \\ \\ Recall \ that: \\ \\ m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \\ So: \\ \\ (x_{1},y_{1})=(2,140) \rightarrow Because \ the \ image \ of \ 2 \ is \ 140 \\ \\ (x_{2},y_{2})=(4,280) \rightarrow Because \ the \ image \ of \ 4 \ is \ 280 \\ \\ Accordingly: \\ \\ \boxed{m=\frac{280-140}{4-2}=\frac{70}{1}} \\ \\ or \ simply: \\ \\ m=70[/tex]
Since this line passes through the origin, then:
[tex]y=mx \\ \\ \boxed{y=70x}[/tex]
Here you have another example of proportional relationships. In this case, the number of gallons of white paint is related to the number of gallons of red paint. Since this line passes through the origin, we just need to find the slope in order to build up the equation of the line:
[tex]Two \ points: \\ \\ (x_{1},y_{1})=(0,0) \\ \\ (x_{2},y_{2})=(4,1) \\ \\ m=\frac{1-0}{4-0} \\ \\ m=\frac{1}{4}[/tex]
Finally, the equation of the line is:
[tex]\boxed{y=\frac{1}{4}x}[/tex]
His graph is incorrect. In fact, the graph comes from the equation:
[tex]y=x[/tex]
Remember that we said above that if m < 0, the line decreases its outputs if you move from left to right.
In the graph occurs the opposite because the line increases its outputs if you move from left to right or, in other words, m > 0 and in this case m = 1, so:
[tex]y=x[/tex]
Equation of lines: https://brainly.com/question/12169569
Proportional relationships: https://brainly.com/question/674693
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