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 perpendicular Yes…these lines are perpendicular No…these lines are parallel No…these lines are parallel No…these lines are perpendicular No…these lines are perpendicular You cannot tell without a graph You cannot tell without a graph Yes…these lines are not parallel Yes…these lines are not parallel Yes…these lines are parallel
No…these lines are parallel
To determine if the line through (3, 7) and (-2, -5) intersects with the line through (4, 8) and (10, -2), we can calculate the slopes of the two lines.
The slope of the line through (3, 7) and (-2, -5) can be found using the formula:
m1 = (y2 - y1) / (x2 - x1)
m1 = (-5 - 7) / (-2 - 3)
m1 = -12 / -5
m1 = 12/5
The slope of the line through (4, 8) and (10, -2) can be found using the same formula:
m2 = (-2 - 8) / (10 - 4)
m2 = -10 / 6
m2 = -5/3
Since the slopes of the two lines, 12/5 and -5/3, are not equal, we conclude that the lines are not parallel.
Therefore, the correct response is: Yes... these lines are not parallel.
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) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. 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? Skip to navigation
To determine if the line through (3, 7) and (-2, -5) intersects with the line through (4, 8) and (10, -2), we can start by finding the equations of both lines using the point-slope formula.
For the first line:
Step 1: Find the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Let's choose (3, 7) as (x1, y1) and (-2, -5) as (x2, y2).
m = (-5 - 7) / (-2 - 3) = -12 / -5 = 12/5
Step 2: Use the slope-intercept form of a line (y = mx + b) and substitute one of the points to find the y-intercept (b). We'll use (3, 7).
7 = (12/5)(3) + b
7 = 36/5 + b
b = 35/5 - 36/5 = -1/5
So the equation of the first line is y = (12/5)x - 1/5.
For the second line:
Follow the same steps using (4, 8) and (10, -2).
Step 1: m = (-2 - 8) / (10 - 4) = -10 / 6 = -5/ 3
Step 2: Using the slope-intercept form and substituting (4, 8):
8 = (-5/3)(4) + b
8 = -20/3 + b
b = 24/3 + 20/3 = 44/3
So the equation of the second line is y = (-5/3)x + 44/3.
Now let's compare the slopes of the two lines. If the slopes are not equal, the lines are not parallel, and therefore intersect. If the slopes are equal, the lines could either be parallel or perpendicular, so we need to compare the slopes and the y-intercepts.
The first line has a slope of 12/5 and a y-intercept of -1/5.
The second line has a slope of -5/3 and a y-intercept of 44/3.
Since the slopes are not equal, the lines are not parallel.
Therefore, the correct response is: Yes...these lines are not parallel.
To determine which option is best in each scenario, we will compare the total cost of each option.
1) You plan to attend 3 times this summer:
The cost of option 1 (individual visits) = 3 * $28 = $<<3*28=84>>84
The cost of option 2 (monthly membership + per visit) = $90 + (3 * $15) = $<<90+(3*15)=135>>135
Therefore, the best option is option 1 (individual visits) as it costs $84, which is less than the cost of option 2.
2) You think you will visit approximately 8 times this summer:
The cost of option 1 (individual visits) = 8 * $28 = $<<8*28=224>>224
The cost of option 2 (monthly membership + per visit) = $90 + (8 * $15) = $<<90+(8*15)=210>>210
Therefore, the best option is option 2 (monthly membership + per visit) as it costs $210, which is less than the cost of option 1.
3) You have budgeted $150 for visiting the park this summer:
We will compare the costs of both options to see which is within your budget.
The maximum number of visits you can make with option 1 (individual visits) is $150 / $28 = 5.36 visits (approximately).
The cost of these 5 visits is 5 * $28 = $<<5*28=140>>140, which is within your budget. Therefore, option 1 is the best choice.
The maximum number of visits you can make with option 2 (monthly membership + per visit) is ($150 - $90) / $15 = 4 visits.
The cost of these 4 visits is 4 * $15 = $<<4*15=60>>60, which is within your budget. Therefore, option 2 is the best choice.
4) To find the break-even point where both options cost about the same, we can set up the equation:
$90 + ($15 * x) = $28 * x
Where x is the number of visits. We can solve this equation to find the value of x.
$90 + $15x = $28x
$90 = $28x - $15x
$90 = $13x
x = $90 / $13
x ≈ 6.92
Therefore, the approximate break-even point is when you make approximately 7 visits.
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?
As an AI language model, I don't have personal preferences. However, I can provide information about different methods for solving a system of linear equations.
There are several methods for solving a system of linear equations, including:
1) Graphing method: This involves graphing each equation on the coordinate plane and finding the point of intersection. This method is useful for visualizing the solutions and can be straightforward when the equations are simple. However, it may not be precise for solving large systems of equations or when the solutions are not easily determined from the graph.
2) Substitution method: In this method, one equation is solved for one variable, and the resulting expression is substituted into the other equation. This allows for the determination of the value of the remaining variable, leading to the solution. This method is useful when one or both equations can be easily solved for a single variable. However, it can become complicated when dealing with complex equations or systems with multiple variables.
3) Elimination method (also known as the addition or subtraction method): This involves manipulating the equations by adding or subtracting them to eliminate one variable. This process continues until one variable can be solved, and the value is substituted back into one of the original equations to find the remaining variable. This method is efficient when both equations are in standard form and can be easily manipulated. However, it can be time-consuming and prone to errors in systems with numerous variables or complex coefficients.
The choice of the method depends on various factors, such as the complexity of the equations, the number of variables, and personal preference. Some may find substitution method easier if they are comfortable with solving simple equations, while others may prefer elimination method for its systematic approach.
It is always helpful to practice and become familiar with multiple methods so that you can choose the most suitable one based on the given system of equations and your comfort level with each method.