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Vladimir79 [104]

3 years ago

14

During their last game, the Miami Dolphins scored ͸ times for a total score of ͵Ͳ points. They scored ͹ points for each touchdow

n and ͵ points for each field goal. Write and solve the systems of equations to find the total touchdowns and field goals scored.

1 answer:

Karolina [17]3 years ago

4 0

Answer:

The Miami Dolphins had 3 touchdowns and 3 field goals.

Step-by-step explanation:

It is given that During their last game, the Miami Dolphins scored 6 times, for a total score of 30 points. They scored 7 points for each touchdown and 3 points for each field goal.

We need to write and solve the system of equations to find the total touchdowns and field goals scored.

Let x represents the number of touchdowns.

Let y be the number of field goals.

It is given that the Miami Dolphins scored 6 times

$x+y=6$

$x=6-y$ ....(1)

They scored 7 points for each touchdown and 3 points for each field goal.

The total score is 30 points:

$7x+3y=30$ ...(2)

Put the value of x from equation (1) into equation (2).

$7( 6-y)+3y=30$

$42-7y+3y=30$

$-4y=30-42$

$-4y=-12$

$y=3$

Put y = 3 in equation 1.

$x=6-3$

$x=3$

Hence, the Miami Dolphins had 3 touchdowns and 3 field goals.

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We are given system of equations

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Let a matrix A= $\left[\begin{array}{ccc}5&4&5\\1&1&2\\2&1&-1\end{array}\right]$ and B= $\left[\begin{array}{ccc}-1\\1\\-3\end{array}\right]$

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$\left[\begin{array}{ccc}1&0&1\\1&1&2\\0&-1&-3\end{array}\right]$ : $\left[\begin{array}{ccc}-5\\1\\-5\end{array}\right]$

Apply $R_2\rightarrow R_2-R_1$

$\left[\begin{array}{ccc}1&0&1\\0&1&1\\0&-1&-3\end{array}\right]$ : $\left[\begin{array}{ccc}-5\\6\\-5\end{array}\right]$

Apply $R_3\rightarrow R_3+R_2$

$\left[\begin{array}{ccc}1&0&1\\0&1&1\\0&0&-2\end{array}\right]$ : $\left[\begin{array}{ccc}-5\\6\\1\end{array}\right]$

Apply $R_3\rightarrow- \frac{1}{2}$ and $R_2\rightarrow R_2-R_3$

$\left[\begin{array}{ccc}1&0&1\\0&1&0\\0&0&1\end{array}\right]$ : $\left[\begin{array}{ccc}-5\\\frac{13}{2}\\-\frac{1}{2}\end{array}\right]$

Apply $R_1\rightarrow R_1-R_3$

$\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]$ : $\left[\begin{array}{ccc}-\frac{9}{2}\\\frac{13}{2}\\-\frac{1}{2}\end{array}\right]$

Rank of matrix A and B are equal.Therefore, matrix A has infinite number of solutions.

Therefore, rank of matrix is equal to rank of B.

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