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Consider the following system of equations:

y−2x=−6
2y = x+3


A) Is (7,8)
a solution to the system of equations? Show your work to justify your answer.

(1 point)
Responses

Yes
Yes

No

5 answers

To determine if the point \( (7, 8) \) is a solution to the system of equations, we need to substitute \( x = 7 \) and \( y = 8 \) into both equations and check if they hold true.

The system of equations is:

  1. \( y - 2x = -6 \)
  2. \( 2y = x + 3 \)

Step 1: Substitute into the first equation. \[ y - 2x = -6 \] Substituting \( y = 8 \) and \( x = 7 \): \[ 8 - 2(7) = -6 \] Simplifying this: \[ 8 - 14 = -6 \] \[ -6 = -6 \quad \text{(True)} \]

Step 2: Substitute into the second equation. \[ 2y = x + 3 \] Substituting \( y = 8 \) and \( x = 7 \): \[ 2(8) = 7 + 3 \] Simplifying this: \[ 16 = 10 \quad \text{(False)} \]

Since the second equation does not hold true, the point \( (7, 8) \) is not a solution to the system of equations.

The answer is: No.

Solve the system of equations using the elimination method. Show your work on a piece of paper.

Type the ordered pair(s) (x,y)
that are the solution(s) to the system of equations

To solve the system of equations using the elimination method, we will first write down the equations. The given equations are:

  1. \( y - 2x = -6 \)
  2. \( 2y = x + 3 \)

Step 1: Rewrite the equations in a standard form.

Starting with the first equation: \[ y - 2x = -6 \quad \Rightarrow \quad 2x + y = -6 \quad (1) \]

Now the second equation: \[ 2y = x + 3 \quad \Rightarrow \quad -x + 2y = 3 \quad (2) \]

So, we have:

  1. \( 2x + y = -6 \)
  2. \( -x + 2y = 3 \)

Step 2: Multiply the equations to align coefficients for elimination.

Let's eliminate \( y \) by matching the coefficients. We can multiply the first equation by 2 to align the \( y \) terms.

Multiplying the first equation by 2: \[ 2(2x + y) = 2(-6) \quad \Rightarrow \quad 4x + 2y = -12 \quad (3) \]

Now we have:

  1. \( 4x + 2y = -12 \quad (3) \)
  2. \( -x + 2y = 3 \quad (2) \)

Step 3: Eliminate \( y \) by subtracting the second equation from the first equation.

Now, we subtract equation (2) from equation (3): \[ (4x + 2y) - (-x + 2y) = -12 - 3 \] This simplifies to: \[ 4x + 2y + x - 2y = -12 - 3 \] \[ 5x = -15 \]

Step 4: Solve for \( x \). \[ x = \frac{-15}{5} = -3 \]

Step 5: Substitute \( x \) back into one of the original equations to find \( y \).

Using equation (1): \[ 2(-3) + y = -6 \] \[ -6 + y = -6 \] \[ y = 0 \]

Final Solution: The solution to the system of equations is: \[ (x, y) = (-3, 0) \]

So, the ordered pair is: \[ \boxed{(-3, 0)} \]

Victor goes to the fair for the rides. The fair in his town is small, so it only has two rides. The ferris wheel costs $2.50 to ride and the ride lasts 7 minutes. The merry-go-round costs $3.00 to ride and the ride lasts 4 minutes. He has $30 and one hour to ride rides. How many times can he ride each ride?



[To type inequality symbols ≤
or ≥
, type <= or >=.]



A) Write two linear inequalities that represent this situation. Let f
be the number of times he rides the ferris wheel, and m
be the number of times he rides the merry-go-round.

(4 points)
Write one inequality about the cost of the rides:

Write one inequality about how long it takes to ride them:

Question 2
B) Can Victor ride the ferris wheel 7 times and the merry-go-round 4 times? Show your work to justify your answer.(1 point)
Responses

Yes
Yes

No

To model the situation described with Victor going to the fair, we will create two linear inequalities representing the constraints on the number of rides based on cost and time.

A) Write the inequalities.

Let:

  • \( f \) = number of times Victor rides the ferris wheel
  • \( m \) = number of times Victor rides the merry-go-round

  1. Cost Inequality: The cost of the ferris wheel ride is $2.50, and the cost of the merry-go-round is $3.00. Victor has $30 to spend.

    The inequality representing the cost is: \[ 2.50f + 3.00m \leq 30 \]

  2. Time Inequality: The ferris wheel ride lasts 7 minutes, and the merry-go-round ride lasts 4 minutes. Victor has 1 hour (or 60 minutes) to spend on rides.

    The inequality representing the time is: \[ 7f + 4m \leq 60 \]

B) Can Victor ride the ferris wheel 7 times and the merry-go-round 4 times?

Let's substitute \( f = 7 \) and \( m = 4 \) into both inequalities to check if they hold true.

  1. Check the cost inequality: \[ 2.50(7) + 3.00(4) \leq 30 \] Calculate the left side: \[ 17.50 + 12.00 = 29.50 \] Now, check if \( 29.50 \leq 30 \) is true. It is true.

  2. Check the time inequality: \[ 7(7) + 4(4) \leq 60 \] Calculate the left side: \[ 49 + 16 = 65 \] Now, check if \( 65 \leq 60 \) is true. It is false.

Conclusion:

Since the time inequality does not hold true, the answer is:

No, Victor cannot ride the ferris wheel 7 times and the merry-go-round 4 times.