How can you tell the amount of solutions a system of equations has??
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Number of solutions to system of equations review
One solution. A system of linear equations has one solution when the graphs intersect at a point.
No solution. A system of linear equations has no solution when the graphs are parallel.
Infinite solutions. A system of linear equations has infinite solutions when the graphs are the exact same line.
Step-by-step explanation:
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To write the system we need the slope of each line and at least one point on the line. The two lines to consider will be the lines connecting the location of each plane to the airport they are flying to. It is also worth noting that the coordinates of the airport represent the point of intersection of the two lines and thus the solution to the system.
1. slope of the line connecting airplane one and the airport: m = 2 you can see this clearly if you graph the two points. From airplane 1 location we rise 8 units and move to the right 4 units to get to the airport. Slope is defined as rise over run: so 8 divided by 4 = 2(the slope) Now substitute the slope and the point (2,4) into point-slope form of a line:
y - 4 = 2(x -4) the standard form of this equation is 2x - y = 0
2. slope of the line connecting airplane 2 and the airport: m = -
To find this slope, simply observe the vertical change of down 3 and a horizontal shift of right 9 from the airport to airplane 2. Now substitute this slope and and the point (15,9) into point-slope form of a line:
y - 9 =
(x - 15) the standard form of this equation is:
x + 3y = 42
Let's write the system:
2x - y = 0
x + 3y = 42
Multiply the first equation by 3 to get the new system
6x - 3y = 0
x + 3y = 42 add these two equations to get an equation in terms of x
7x = 42 thus x = 6 and substituting this value into 2x - y = 0 we see y = 12
In other words, we have proven that the location of the airport is in fact the solution to our system.
PS: You just have to do a little algebra to get from point-slope form of the two equations to standard form. I did not show this process, but if you need it just let me know... thanks
1. slope of the line connecting airplane one and the airport: m = 2 you can see this clearly if you graph the two points. From airplane 1 location we rise 8 units and move to the right 4 units to get to the airport. Slope is defined as rise over run: so 8 divided by 4 = 2(the slope) Now substitute the slope and the point (2,4) into point-slope form of a line:
y - 4 = 2(x -4) the standard form of this equation is 2x - y = 0
2. slope of the line connecting airplane 2 and the airport: m = -
y - 9 =
x + 3y = 42
Let's write the system:
2x - y = 0
x + 3y = 42
Multiply the first equation by 3 to get the new system
6x - 3y = 0
x + 3y = 42 add these two equations to get an equation in terms of x
7x = 42 thus x = 6 and substituting this value into 2x - y = 0 we see y = 12
In other words, we have proven that the location of the airport is in fact the solution to our system.
PS: You just have to do a little algebra to get from point-slope form of the two equations to standard form. I did not show this process, but if you need it just let me know... thanks
Answer:
<h2>See the explanation.</h2>Step-by-step explanation:
a.
The initial length of the candle is 16 inch. It also given that, it burns with a constant rate of 0.8 inch per hour.
After one hour since the candle was lit, the length of the candle will be (16 - 0.8) = 15.2 inch.
After two hour since the candle was lit, the length of the candle will be (15.2 - 0.8) = 14.4 inch. The length of the candle after two hours can also be represented by {16 - 2(0.8)}.
Hence, the length of the candle after t hours when it was lit can be represented by the function, .
at t = 20.
b.
The domain of the function is 0 to 20.
c.
The range is 0 to 16.