Solve the system of inequalities by graphing.

To solve the system of inequalities by graphing, we will follow these steps:

1. Graph each inequality.
2. Determine the type of line (solid or dotted).
3. Shade the appropriate regions.

Step 1: Graph the inequalities

Inequality 1: \( y < 10x - 4 \)

• To graph the line \( y = 10x - 4 \):

  • Find the y-intercept (when \( x = 0 \)):
    • \( y = 10(0) - 4 = -4 \). So, the point is \( (0, -4) \).
  • Find another point (choose \( x = 1 \)):
    • \( y = 10(1) - 4 = 6 \). So, the point is \( (1, 6) \).

• Draw a dotted line because the inequality is \( < \), indicating that points on the line are not included in the solution set.

Inequality 2: \( y \geq \frac{1}{4}x - 3 \)

• To graph the line \( y = \frac{1}{4}x - 3 \):

  • Find the y-intercept (when \( x = 0 \)):
    • \( y = \frac{1}{4}(0) - 3 = -3 \). So, the point is \( (0, -3) \).
  • Find another point (choose \( x = 4 \)):
    • \( y = \frac{1}{4}(4) - 3 = 1 - 3 = -2 \). So, the point is \( (4, -2) \).

• Draw a solid line because the inequality is \( \geq \), indicating that points on the line are included in the solution set.

Step 2: Shade the regions

• For the inequality \( y < 10x - 4 \): Shade below the dotted line.
• For the inequality \( y \geq \frac{1}{4}x - 3 \): Shade above the solid line.

Step 3: Find the solution region

The solution to the system of inequalities is the region where the shaded areas overlap.

Summary

1. Graph the line \( y = 10x - 4 \) with a dotted line and shade below.
2. Graph the line \( y = \frac{1}{4}x - 3 \) with a solid line and shade above.
3. The solution region is the area where both shadings intersect.

Visualization

You'll need to use graphing software or draw it manually on graph paper to see the exact shaded regions. Make sure to extend the graph to show both axes up to 10, as noted.

For his parents' anniversary party, Lee is considering using one of two venues. A hotel in Clarksville will cost $500 for a reservation, plus $8 per person. A restaurant in the same city will cost $6 per person, in addition to $600 for the reservation. In order to make the best decision, Lee figures out how many attendees it would take to have the venues cost the same amount. How many attendees would that be?
Write a system of equations, graph them, and type the solution.

To find how many attendees it would take for the costs of the two venues to be the same, we can set up equations based on the given costs.

Step 1: Set up the equations

1. Hotel Costs:

   • The total cost for the hotel in Clarksville can be expressed as: \[ C_h = 500 + 8x \] where \( C_h \) is the cost for the hotel and \( x \) is the number of attendees.

2. Restaurant Costs:

   • The total cost for the restaurant can be expressed as: \[ C_r = 600 + 6x \] where \( C_r \) is the cost for the restaurant.

Step 2: Set up the system of equations

To find out when the costs are equal, we set the two equations equal to each other: \[ 500 + 8x = 600 + 6x \]

Step 3: Solve for \( x \)

1. Rearranging the equation gives: \[ 500 + 8x - 6x = 600 \] \[ 500 + 2x = 600 \]
2. Subtract 500 from both sides: \[ 2x = 100 \]
3. Divide by 2: \[ x = 50 \]

Conclusion

The number of attendees for the costs to be the same is \( 50 \).

Step 4: Graph the equations

To visualize this, we can graph the two cost equations.

1. Hotel Cost Equation: \( C_h = 500 + 8x \)
2. Restaurant Cost Equation: \( C_r = 600 + 6x \)

Both equations can be rearranged into slope-intercept form \( y = mx + b \):

1. Hotel: \[ C_h = 8x + 500 \]

2. Restaurant: \[ C_r = 6x + 600 \]

Step 5: Intersection Point

Now, you graph both equations on the same set of axes:

• For the Hotel:

  • When \( x = 0 \), \( C_h = 500 \) (point (0, 500)).
  • When \( x = 50 \), \( C_h = 500 + 8(50) = 500 + 400 = 900 \) (point (50, 900)).

• For the Restaurant:

  • When \( x = 0 \), \( C_r = 600 \) (point (0, 600)).
  • When \( x = 50 \), \( C_r = 600 + 6(50) = 600 + 300 = 900 \) (point (50, 900)).

The lines intersect at the point \( (50, 900) \).

Solution

• Number of attendees for equal cost: 50 attendees
• Total cost: $900 for both venues when 50 guests attend.

You can use graphing software to plot the two lines, and you will see that they intersect at the point \((50, 900)\).

Is (–3, –9) a solution to this system of inequalities?

y < –3x − 8
y ≤ 2x − 3

yes or no

To determine whether the point \((-3, -9)\) is a solution to the system of inequalities, we need to substitute \(x = -3\) and \(y = -9\) into both inequalities and check if they hold true.

First Inequality: \(y < -3x - 8\)

Substituting \((-3, -9)\) into the first inequality:

\[ -9 < -3(-3) - 8 \]

Calculating the right side:

\[ -9 < 9 - 8 \] \[ -9 < 1 \]

This statement is true.

Second Inequality: \(y \leq 2x - 3\)

Substituting \((-3, -9)\) into the second inequality:

\[ -9 \leq 2(-3) - 3 \]

Calculating the right side:

\[ -9 \leq -6 - 3 \] \[ -9 \leq -9 \]

This statement is true.

Conclusion

Since both inequalities are satisfied by the point \((-3, -9)\), the answer is yes; \((-3, -9)\) is a solution to the system of inequalities.

Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

Hayley received some gift cards for music and movie downloads for her birthday. Using one of them, she downloaded 20 songs and 20 movies, which cost a total of $240. Using another, she purchased 19 songs and 13 movies, which cost a total of $168. How much does each download cost?

Downloads cost $(_______) for a song and $(______)for a movie.

To write a system of equations based on the given situation, let:

• \( x \) = cost of one song
• \( y \) = cost of one movie

Step 1: Set up the equations

From the information provided:

1. First situation: Hayley downloaded 20 songs and 20 movies for a total of $240, which gives the equation: \[ 20x + 20y = 240 \] We can simplify this by dividing everything by 20: \[ x + y = 12 \quad \text{(Equation 1)} \]

2. Second situation: Hayley purchased 19 songs and 13 movies for a total of $168, which gives the equation: \[ 19x + 13y = 168 \quad \text{(Equation 2)} \]

Step 2: Solve the system using elimination

We now have the system of equations:

1. \( x + y = 12 \) (Equation 1)
2. \( 19x + 13y = 168 \) (Equation 2)

To use elimination, we can first express \( y \) from Equation 1:

\[ y = 12 - x \]

Now, substitute this expression for \( y \) in Equation 2:

\[ 19x + 13(12 - x) = 168 \]

Distribute the 13:

\[ 19x + 156 - 13x = 168 \]

Combine like terms:

\[ 6x + 156 = 168 \]

Now, isolate \( x \):

\[ 6x = 168 - 156 \] \[ 6x = 12 \] \[ x = 2 \]

Now that we have \( x \), substitute it back into Equation 1 to find \( y \):

\[ x + y = 12 \] \[ 2 + y = 12 \] \[ y = 12 - 2 \] \[ y = 10 \]

Conclusion

The costs for each download are:

• The cost of a song is $2.
• The cost of a movie is $10.

Therefore, the final answer is:

• Downloads cost \($ (2)\) for a song and \($ (10)\) for a movie.

Is (2,7) a solution to this system of equations?
y=3x+1
y=2x+5
yes or no

To determine whether the point \((2, 7)\) is a solution to the system of equations, we need to substitute \(x = 2\) and \(y = 7\) into each equation and see if they hold true.

Check the First Equation: \(y = 3x + 1\)

Substituting \((2, 7)\):

\[ y = 3x + 1 \] \[ 7 = 3(2) + 1 \] \[ 7 = 6 + 1 \] \[ 7 = 7 \quad \text{(True)} \]

Check the Second Equation: \(y = 2x + 5\)

Substituting \((2, 7)\):

\[ y = 2x + 5 \] \[ 7 = 2(2) + 5 \] \[ 7 = 4 + 5 \] \[ 7 = 9 \quad \text{(False)} \]

Conclusion

Since the point \((2, 7)\) satisfies the first equation but not the second equation, it is not a solution to the system of equations.

The answer is no.