Questions: Use the system of inequalities to answer the following question: y > -2 x + 3, y ≥ x - 4. What is the x-value of the intersection of the boundary lines? A. -7/3 B. -3/7 C. 3/7 D. 7/3 A biologist is developing two new strains of bacteria. Each sample of Type I produces four new viable bacteria, and each sample of Type II produces three new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, but no more than 60, of the original samples must be Type I. Not more than 70 of the samples can be Type II. A sample of Type I costs 5 and a sample of Type II costs 7. How many samples of Type II bacteria should be used to minimize cost? A. 0 B. 30 C. 60

Use the system of inequalities to answer the following question: y > -2 x + 3, y ≥ x - 4. What is the x-value of the intersection of the boundary lines? A. -7/3 B. -3/7 C. 3/7 D. 7/3 A biologist is developing two new strains of bacteria. Each sample of Type I produces four new viable bacteria, and each sample of Type II produces three new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, but no more than 60, of the original samples must be Type I. Not more than 70 of the samples can be Type II. A sample of Type I costs 5 and a sample of Type II costs 7. How many samples of Type II bacteria should be used to minimize cost? A. 0 B. 30 C. 60
Transcript text: 12. Use the system of inequalities to answer the following question: $\left\{\begin{array}{c}y>-2 x+3 \\ y \geq x-4\end{array}\right.$. What is the $x$-value of the intersection of the boundary lines? A. $-\frac{7}{3}$ B. $-\frac{3}{7}$ C. $\frac{3}{7}$ D. $\frac{7}{3}$ 13. A biologist is developing two new strains of bacteria. Each sample of Type I produces four new viable bacteria, and each sample of Type II produces three new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, but no more than 60, of the original samples must be Type I. Not more than 70 of the samples can be Type II. A sample of Type I costs $\$ 5$ and a sample of Type II costs $\$ 7$. How many samples of Type II bacteria should be used to minimize cost? A. 0 B. 30 C. 60
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Solution

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Solution Steps

Step 1: Finding the Intersection of the Boundary Lines

To find the intersection of the boundary lines defined by the inequalities \( y > -2x + 3 \) and \( y \geq x - 4 \), we first convert these equations into standard form:

  1. From \( y = -2x + 3 \), we rearrange to get \( 2x + y = 3 \).
  2. From \( y = x - 4 \), we rearrange to get \( -x + y = -4 \).

We can represent this system of equations in matrix form as follows:

\[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 2 & 1 & 3 \\ -1 & 1 & -4 \\ \end{array} \right] \]

After applying Gaussian elimination, we arrive at the reduced row echelon form:

\[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & 0 & \frac{7}{3} \\ 0 & 1 & -\frac{5}{3} \\ \end{array} \right] \]

This gives us the solution:

\[ x = \frac{7}{3}, \quad y = -\frac{5}{3} \]

Step 2: Minimizing the Cost of Bacteria Samples

The biologist needs to produce at least 240 new viable bacteria using two types of samples. The constraints are:

  • Each Type I sample produces 4 bacteria.
  • Each Type II sample produces 3 bacteria.
  • The cost for Type I is \$5 and for Type II is \$7.
  • The number of Type I samples \( x \) must satisfy \( 30 \leq x \leq 60 \).
  • The number of Type II samples \( y \) must satisfy \( 0 \leq y \leq 70 \).

The cost function to minimize is:

\[ C = 5x + 7y \]

We evaluate the cost at the boundary points of the feasible region:

  1. At \( (30, 70) \): \( C = 5(30) + 7(70) = 640 \)
  2. At \( (60, 40) \): \( C = 5(60) + 7(40) = 580 \)
  3. At \( (30, 50) \): \( C = 5(30) + 7(50) = 500 \)
  4. At \( (60, 0) \): \( C = 5(60) + 7(0) = 300 \)

The minimum cost occurs at the point \( (60, 0) \) with a cost of \$300.

Final Answer

The \( x \)-value of the intersection of the boundary lines is \( \boxed{\frac{7}{3}} \).
The number of Type II samples to minimize cost is \( \boxed{0} \).

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