Which graph represents the proportional relationship given by the rule "y is 3 times the value of x"?
2 answers:
Answer:
The fourth graph i.e. RIGHT BOTTOM GRAPH is correct which represents the proportional relationship given by the rule y = 3x.
Also, the correct graph is attached below.
Step-by-step explanation:
Given the rule
''y is 3 times the value of x".
It means
y = 3x
comparing with the slope-intercept form of the line equation
y = mx+b
where y is the slope and b is the intercept
so,
The slope of y = 3x will be: m = 3
The y-intercept of y = 3x will be: b = 0
CHECKING THE 4TH GRAPH (THE RIGHT BOTTOM SIDE)
Taking two points
(0, 0)
(2, 6)
Finding the slope between (0, 0) and (2, 6)
Using the formula
Slope = m = [y₂ - y₁] / [x₂ - x₁]
= [6 - 0] / [2 - 0]
= 6 / 2
= 3
Thus, the slope of the line = m = 3
We know that the value of y-intercept can be determined by setting x = 0, and determining the corresponding value of y.
From the graph, it is clear
at x = 0, y = 0
Thus, the y-intercept b = 0
Now, substituting m = 3 and b = 0 in the slope-intercept form of linear function
y = mx+b
y = 3x+0
y = 3x
Therefore, the fourth graph i.e. RIGHT BOTTOM GRAPH is correct which represents the proportional relationship given by the rule y = 3x.
Also, the correct graph is attached below.
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2 Answers:
- B) The lines are parallel
- C) The lines have the same slope.
Parallel lines always have equal slope, but different y intercepts.
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Explanation:
Let's solve the second equation for y
3y - x = -7
3y = -7+x
3y = x-7
y = (x-7)/3
y = x/3 - 7/3
y = (1/3)x - 7/3
The equation is in y = mx+b form with m = 1/3 as the slope and b = -7/3 as the y intercept. We see that the first equation, where y was already isolated, also has a slope of m = 1/3. The two equations of this system have the same slope. Choice C is one of the answers.
However, they don't have the same y intercept. The first equation has y intercept b = -4, while the second has b = -7/3. This means that they do not represent the same line. They need to have identical slopes, and identical y intercepts (though the slope can be different from the y intercept of course) in order to have identical lines. So we can rule out choice D and E because of this.
Because the two equations have the same slope, but different y intercepts, this means the lines are parallel. Choice B is the other answer.
Parallel lines never touch or intersect, which in turn means there is no solution point. A solution point is where the lines cross. We can rule out choice A.
I recommend using your graphing calculator, Desmos, GeoGebra, or any graphing tool (on your computer or online) to graph each equation given. You should see two parallel lines forming. I used GeoGebra to make the graph shown below.
