Solve the following systems of equations algebraically (first) and graphically: A. Melissa and Trevor sell candy bars to raise money for a class field trip. Trevor sells 3 more than 6 times as many candy bars as Melissa sells. Together they sell 24 candy bars. Let m represent the number of candy bars Melissa sells. Let t represent the number of candy bars Trevor sells. Write a system of linear equations to represent this situation. using elimination and x and y variables
We are given the following information:
1) Trevor sells 3 more than 6 times as many candy bars as Melissa sells:
t = 6m + 3
2) Together they sell 24 candy bars:
m + t = 24
To solve this system of equations algebraically using elimination, we can substitute the expression for t from equation 1 into equation 2:
m + (6m + 3) = 24
Simplifying this equation will give us the value for m, which we can then plug back into equation 1 to find the value for t.
Let's solve it:
m + 6m + 3 = 24
7m + 3 = 24
7m = 21
m = 3
Now, we can substitute the value of m back into equation 1 to find the value for t:
t = 6(3) + 3
t = 18 + 3
t = 21
Therefore, Melissa sells 3 candy bars (m = 3) and Trevor sells 21 candy bars (t = 21).
To graphically represent this system of equations, we can plot the equations on a coordinate grid.
First, let's rearrange equation 2 to solve for t:
t = 24 - m
Now, let's create a table of values for m and t:
For m = 0, t = 24 - 0 = 24.
For m = 1, t = 24 - 1 = 23.
For m = 2, t = 24 - 2 = 22.
For m = 3, t = 24 - 3 = 21.
For m = 4, t = 24 - 4 = 20.
Plotting these points on the coordinate grid, we get:
(
0, 24
)
(
1, 23
)
(
2, 22
)
(
3, 21
)
(
4, 20
)
Now, let's plot the first equation:
t = 6m + 3.
Choosing a few values for m and finding the corresponding values for t, we have:
For m = 0, t = 3.
For m = 1, t = 6 + 3 = 9.
For m = 2, t = 12 + 3 = 15.
For m = 3, t = 18 + 3 = 21.
For m = 4, t = 24 + 3 = 27.
Plotting these points on the coordinate grid, we get:
(
0, 3
)
(
1, 9
)
(
2, 15
)
(
3, 21
)
(
4, 27
)
We can see that the two graphs intersect at the point (3, 21). This corresponds to the solution of the system of equations, m = 3 and t = 21.