Question

How many solutions do the problems have?

1.
y = 3x + 6
y = -13x - 4
2.
x - 2y = 63x - 6
y = 18
3.
y = 23x - 1
y = 23x - 2

Answer 1

[AI-generated answer — Bot GPT 3.5]

1. The equations y = 3x + 6 and y = -13x - 4 form a system of two linear equations. Since the slopes of the two equations (3 and -13) are not equal, the system has one unique solution.

2. The equations x - 2y = 63x - 6 and y = 18 form a system of two linear equations. Substituting y = 18 into the first equation, we get x - 2(18) = 63x - 6, which simplifies to x - 36 = 63x - 6. Rearranging the terms, 0x - 62x = 6 - 36, or -62x = -30. Solving for x, we get x = 30/62 = 15/31. Substituting this value of x into y = 18, we find y = 18. Therefore, the system has one unique solution.

3. The equations y = 23x - 1 and y = 23x - 2 form a system of two linear equations. Since the slopes of the two equations are equal (both 23), the lines are parallel and will never intersect. Therefore, the system has no solution.

Answer 2

[AI-generated answer — Step-by-Step Bot]

Let's analyze each problem step-by-step to find the number of solutions:

1.
The given equations are:
y = 3x + 6
y = -13x - 4

To determine the number of solutions, we need to compare the slopes of the two lines.

In this case, the slopes of the lines are 3 and -13 respectively. Since the slopes are different, these lines will intersect at a single point, which means they have one solution.

2.
The given equations are:
x - 2y = 6
y = 18

In this case, the second equation is a horizontal line and doesn't involve x. This implies that y always equals 18, regardless of the value of x.

When we substitute y=18 into the first equation:
x - 2(18) = 6
x - 36 = 6
x = 6 + 36
x = 42

So, this system of equations has a unique solution (x=42, y=18).

3.
The given equations are:
y = 23x - 1
y = 23x - 2

In this case, if we subtract the second equation from the first equation, we get:
(23x - 1) - (23x - 2) = 0
23x - 1 - 23x + 2 = 0
1 + 2 = 0

The equation simplifies to 3 = 0, which is not possible. This means that the system of equations has no solution.

In summary:
1. The first system of equations has one solution.
2. The second system of equations has one solution.
3. The third system of equations has no solution.

Answer 3

[AI-generated answer — Explain Bot]

To determine the number of solutions for each problem, we need to compare the given equations.

For Problem 1:
y = 3x + 6
y = -13x - 4

The first equation has a positive coefficient for x, and the second equation has a negative coefficient for x. Since the coefficients are different, we can conclude that the lines represented by these equations will intersect at a single point. Therefore, Problem 1 has one solution.

For Problem 2:
x - 2y = 63x - 6
y = 18

In this problem, there are two equations. The first equation can be simplified as -2y = 62x - 6 or y = -31x + 3. Now we can see that the second equation is independent of x. It means that no matter what value x takes, y will always be 18. In this case, the lines represented by these equations are parallel and will not intersect. Therefore, Problem 2 has no solution.

For Problem 3:
y = 23x - 1
y = 23x - 2

In this case, both equations have the same slope (23) and the same y-intercept (-1 and -2). The equations represent the same line. Therefore, Problem 3 has infinitely many solutions since any point on the line is a valid solution.

To determine the number of solutions for a system of equations, you need to compare the coefficients and solve the equations using different methods such as substitution or elimination. By analyzing the coefficients and the relationships between variables, you can deduce how many solutions each problem has.

Answer 4

How could you check to see if the point (5, 6) is the solution to the Linear System of equations?(1 point)

Responses
Substitute 6 in for x and 5 in for y in both of the equations to see if both equations are true.
Substitute 5 in for x and 6 in for y in one of the equations to see if the equation is true.
Substitute 6 in for x and 5 in for y in one of the equations to see if the equation is true.
Substitute 5 in for x and 6 in for y in both of the equations to see if both equations are true.

Answer 5

[AI-generated answer — Bot GPT 3.5]

To check if the point (5, 6) is a solution to the linear system of equations, you would substitute 5 in for x and 6 in for y in both of the equations to see if both equations are true.

Answer 6

You are running a concession stand at a baseball game. You are selling pizza slices and sodas. Each pizza slice sells for $3.50 and each soda sells for $1.50. At the end of the night, you made a total of $800. You sold a total of 344 pizza slices and sodas combined. You must report the number of pizza slices sold and the number of sodas sold. What equations did you use to solve this, where P = the number of pizza slices sold and S = the number of sodas sold? (1 point)

Responses
3.50P + S = 800 and P + 1.50S = 344
3.50S + 1.50P = 344 and P + S = 800
3.50S + 1.50P = 800 and P + S = 344
3.50P + 1.50S = 800 and P + S = 344

Answer 7

[AI-generated answer — Bot GPT 3.5]

The correct equations to solve this problem are:

3.50P + S = 800
P + 1.50S = 344

In the first equation, 3.5P represents the total revenue from selling pizza slices and S represents the revenue from selling sodas. This equation states that the total revenue from both items is $800.

In the second equation, P represents the number of pizza slices sold and 1.50S represents the number of sodas sold. This equation states that the total number of items sold, which is the sum of pizza slices and sodas, is equal to 344.

By solving these two equations simultaneously, you can find the values of P and S that satisfy both equations, representing the number of pizza slices and sodas sold at the concession stand.

Answer 8

Check all that apply. Select the statement(s) that correctly describes the solution to this system of equations:

8x−2y=−4
4x−y=−2
(2 points) Responses:
Solve this system by elimination since it is already in standard form and lined up nicely.
There is exactly one solution to this system of linear equations and it is (2, 0).
There are infinite solutions to this system of linear equations.
These lines are parallel, so there is no solution to this system of linear equations.
There is exactly one solution to this system of linear equations and it is (0, -2).
Solve this system by substitution since one of the variables is defined by the other without having to do any math.

Answer 9

[AI-generated answer — Bot GPT 3.5]

The correct statement(s) that describe the solution to this system of equations are:

- There is exactly one solution to this system of linear equations and it is (2, 0).

The reason behind this response is that the system of equations is consistent and the coefficients of x and y have different values. This means that the lines represented by the equations intersect at a single point, resulting in one solution. The solution is (2, 0) based on the intersection of the lines.

Answer 10

Jim bought some potted plants: Petunias cost $11 per pot and Begonias cost $15 per pot. Jim bought 19 potted plants in all and spent a total of $225 on the plants. Which would be a correct statement? (1 point)

Responses
Jim bought 15 petunias.
Jim bought 4 petunias.
Jim bought 18 begonias.
Jim bought 15 begonias

Answer 11

[AI-generated answer — Bot GPT 3.5]

To find the correct statement, let's set up the equations based on the given information:

Let P represent the number of Petunias purchased.
Let B represent the number of Begonias purchased.

Based on the given information, we know that:
Petunias cost $11 per pot, so the cost of P Petunias is 11P.
Begonias cost $15 per pot, so the cost of B Begonias is 15B.

Jim bought 19 potted plants in total, so P + B = 19 (equation 1).
Jim spent a total of $225 on the plants, so 11P + 15B = 225 (equation 2).

To find the correct statement, we need to solve the system of equations.

By rearranging equation 1, we have P = 19 - B.

Substituting this value of P into equation 2:
11(19 - B) + 15B = 225
209 - 11B + 15B = 225
4B = 16
B = 4

Substituting B = 4 back into equation 1, we find P = 19 - 4 = 15.

Therefore, the correct statement is:
Jim bought 15 Begonias.

Answer 12

Solve the system of equations graphically. You can use your graphing calculator!

y=−2x−13x − y =−7 (1 point)
Responses.
The solution is (-1.6, 2.2 )
The solution is (0, 2)
The solution is (-1.5, 0)
The solution is (-1, -1)

Answer 13

[AI-generated answer — Bot GPT 3.5]

To solve the system of equations graphically, let's input the two equations into a graphing calculator:

1. y = -2x -1
2. -x - y = -7

Graphing these equations will show the point of intersection, which represents the solution.

By graphing the equations, we observe that the point of intersection is approximately (-1.6, 2.2).

Therefore, the correct statement is:

The solution is (-1.6, 2.2).

Answer 14

To solve this equation you could do the following steps:

3x+4y=48
6x+2y=60 (6 points)
1. Choose the best method to solve by looking at the way the equations are set up for you: ___
2. Using the best method, how could you get the x variables to cancel out? ______
3. After you complete step 2, what would your top equation be? ______
4. Line them up and add the equations together to get this equation. _____
5. Solve for y. _____
6. Solve for x. _____

Answer 15

[AI-generated answer — Bot GPT 3.5]

1. Choose the best method to solve by looking at the way the equations are set up for you:

Based on the equations, the best method to solve this system is to use the method of elimination.

2. Using the best method, how could you get the x variables to cancel out?

To cancel out the x variables, we can multiply the first equation by 2 and the second equation by 3. This will make the coefficients of x in the two equations the same:

Equation 1: 6x + 8y = 96
Equation 2: 18x + 6y = 180

3. After you complete step 2, what would your top equation be?

After multiplying the first equation by 2, the top equation would be: 6x + 8y = 96

4. Line them up and add the equations together to get this equation.

Adding the two equations together, we have:
(6x + 8y) + (18x + 6y) = 96 + 180
24x + 14y = 276

5. Solve for y.

To solve for y, we can isolate the y variable in the equation:
24x + 14y = 276
14y = 276 - 24x
y = (276 - 24x) / 14

6. Solve for x.

To solve for x, we need to substitute the value of y from step 5 into either of the original equations and solve for x. Let's substitute it into the first equation:

3x + 4((276 - 24x) / 14) = 48

Simplifying this equation and solving for x will give us the value of x, and we can then substitute it into step 5 to find the value of y.

The steps to solve for y and x are not provided in this answer.

Answer 16

The Johnsons and the Wilsons went to the Burger Barn for dinner. The Johnsons bought 2 burger meals and 5 hot dog meals for $46. The Wilsons bought 5 burger meals and 2 hot dog meals for $52. When you solve, you get a solution of (8, 6). (3 points)

What does the 8 mean in context? ____
What does the 6 mean in context? ____
Who got the better deal? ____

Answer 17

[AI-generated answer — Bot GPT 3.5]

To understand the meaning of the numbers in context, let's analyze the given information:

The Johnsons bought 2 burger meals and 5 hot dog meals for $46.
The Wilsons bought 5 burger meals and 2 hot dog meals for $52.

When we solve the system of equations derived from this information, we obtain the solution (8, 6).

1. What does the 8 mean in context?

In this context, the number 8 represents the number of burger meals purchased. Since the solution is (8, 6), it means that the Johnsons bought 8 burger meals.

2. What does the 6 mean in context?

In this context, the number 6 represents the number of hot dog meals purchased. Since the solution is (8, 6), it means that the Johnsons bought 6 hot dog meals.

3. Who got the better deal?

To determine who got the better deal, we can compare the cost of the meals for each family.

For the Johnsons, 2 burger meals and 5 hot dog meals cost $46.
For the Wilsons, 5 burger meals and 2 hot dog meals cost $52.

Comparing the costs, we can see that the Johnsons got the better deal since they paid less for their meals ($46) compared to the Wilsons ($52).

Answer 18

Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)? (1 point)

Responses.
Yes…these lines are parallel
No…these lines are parallel
You cannot tell without a graph
Yes…these lines are perpendicular
Yes…these lines are not parallel
No…these lines are perpendicular

Answer 19

[AI-generated answer — Bot GPT 3.5]

No... these lines are not parallel.

To determine if the lines intersect, we can calculate the slopes of each line.

The slope of the line through (3, 7) and (-2, -5) is:
slope = (change in y)/(change in x) = (-5 - 7) / (-2 - 3) = -12 / -5 = 12/5

The slope of the line through (4, 8) and (10, -2) is:
slope = (change in y)/(change in x) = (-2 - 8) / (10 - 4) = -10 / 6 = -5/3

Since the slopes of the two lines are different (12/5 and -5/3), the lines are not parallel. Therefore, they have the potential to intersect. However, we cannot definitively determine whether they intersect or not without additional information or graphing the lines.

Answer 20

An amusement park that you are excited to try is $28 per visit. You have the option to purchase a monthly membership for $90 and then pay $15 for each visit. (4 points)

You have plans to attend 3 times this summer. Which option is best? ______
You think you will visit probably about 8 times this summer. Which option is best? ______
You have budgeted $150 for visiting the park this summer. Which option is best? _____
How many visits would be approximately the break even point where both options would cost about the same? _____

Answer 21

[AI-generated answer — Bot GPT 3.5]

To determine which option is best in each scenario, let's analyze the costs for both options:

Costs for Option 1 (Pay-per-visit):
- Each visit costs $28.

Costs for Option 2 (Monthly membership + Pay-per-visit):
- Monthly membership costs $90.
- Each visit costs an additional $15.

1. You have plans to attend 3 times this summer. Which option is best?

For Option 1: 3 visits x $28 = $84
For Option 2: $90 (monthly membership) + 3 visits x $15 = $135

In this case, Option 1 (Pay-per-visit) is the best as it would cost $84 compared to $135 with Option 2.

2. You think you will visit approximately 8 times this summer. Which option is best?

For Option 1: 8 visits x $28 = $224
For Option 2: $90 (monthly membership) + 8 visits x $15 = $210

In this case, Option 2 (Monthly membership + Pay-per-visit) is the best as it would cost $210 compared to $224 with Option 1.

3. You have budgeted $150 for visiting the park this summer. Which option is best?

For Option 1: $28 per visit
For Option 2: $90 (monthly membership) + $15 per visit

We need to determine the number of visits we can afford within the budget of $150 for both options.

For Option 1: $150 / $28 ≈ 5.36 visits
For Option 2: ($150

Answer 22

Which method of solving a system of linear equation do you prefer to use? Why do you find this particular method easier than the other two?(2 points)