Question

1. how many solutions does the system of equations have?

y = -\frac{7}{2}x + 11
7x + 2y = 20
\textcircled{a} 1 solution
\textcircled{b} no solution
\textcircled{c} infinitely many solutions
\textcircled{d} cannot be determined

1. ten mangos and twelve avocadoes cost $23. five mangos and four avocadoes cost $10. how much do one mango and one avocado cost?

$\square$

1. choose the graph that matches the inequality $y > \frac{2}{3}x - 1$.

\textcircled{a} graph
\textcircled{b} graph
\textcircled{c} graph

envision® florida b.e.s.t. algeb

Explanation:

Question 25

Step1: Rewrite the second equation

We have the system:
\( y = -\frac{7}{2}x + 11 \)
\( 7x + 2y = 20 \)

Let's rewrite the second equation in slope - intercept form (\(y=mx + b\)). Solve \(7x + 2y=20\) for \(y\):

Subtract \(7x\) from both sides: \(2y=-7x + 20\)

Divide both sides by 2: \(y=-\frac{7}{2}x + 10\)

Step2: Compare slopes and y - intercepts

The first equation is \(y = -\frac{7}{2}x+11\) with slope \(m_1 = -\frac{7}{2}\) and y - intercept \(b_1 = 11\)

The second equation (after rewriting) is \(y=-\frac{7}{2}x + 10\) with slope \(m_2=-\frac{7}{2}\) and y - intercept \(b_2 = 10\)

Since the slopes of the two lines are equal (\(m_1=m_2 = -\frac{7}{2}\)) and the y - intercepts are different (\(b_1
eq b_2\)), the two lines are parallel. Parallel lines do not intersect, so the system of equations has no solution.

Step1: Define variables

Let the cost of one mango be \(m\) dollars and the cost of one avocado be \(a\) dollars.

From the problem, we can set up the following system of equations:

\(

$$\begin{cases}10m + 12a=23\\5m+4a = 10\end{cases}$$

\)

Step2: Eliminate one variable

Let's multiply the second equation by 2: \(2\times(5m + 4a)=2\times10\)

Which gives \(10m+8a = 20\)

Now we have the two equations:

1. \(10m + 12a=23\)
2. \(10m+8a = 20\)

Subtract the second equation from the first equation:

\((10m + 12a)-(10m + 8a)=23 - 20\)

Simplify the left - hand side: \(10m+12a - 10m - 8a=4a\)

Simplify the right - hand side: \(3\)

So \(4a = 3\), then \(a=\frac{3}{4}=0.75\)

Step3: Find the value of \(m\)

Substitute \(a = 0.75\) into the second equation \(5m+4a = 10\)

\(5m+4\times0.75=10\)

\(5m + 3=10\)

Subtract 3 from both sides: \(5m=10 - 3=7\)

Divide both sides by 5: \(m=\frac{7}{5}=1.4\)

Step4: Find the cost of one mango and one avocado

The cost of one mango and one avocado is \(m + a\)

\(m + a=1.4+0.75 = 2.15\)

Step1: Analyze the inequality \(y>\frac{2}{3}x - 1\)

• The boundary line of the inequality \(y>\frac{2}{3}x - 1\) is the line \(y=\frac{2}{3}x - 1\). Since the inequality is \(y>\frac{2}{3}x - 1\) (not \(y\geq\frac{2}{3}x - 1\)), the boundary line should be a dashed line.
• To determine which side of the line to shade, we can use a test point. A common test point is \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the inequality:

\(0>\frac{2}{3}(0)-1\)

\(0>- 1\), which is true. So we shade the side of the line that contains the point \((0,0)\)

Step2: Analyze the options

• Option A: The boundary line seems to pass through the origin, and the inequality for the boundary line is not \(y=\frac{2}{3}x - 1\), so this is incorrect.
• Option B: The boundary line is solid, but our inequality has a dashed line (since \(y>\frac{2}{3}x - 1\) not \(y\geq\frac{2}{3}x - 1\)), so this is incorrect.
• Option C: The boundary line is \(y=\frac{2}{3}x - 1\) (dashed line), and when we test the point \((0,0)\), \(0>\frac{2}{3}(0)-1\) ( \(0 > - 1\)) is true, and the shading is on the side that contains \((0,0)\).

Answer:

B. no solution

Question 26