Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)?
No…these lines are perpendicular
You cannot tell without a graph
No…these lines are parallel
Yes…these lines are not parallel
Yes…these lines are parallel
Yes…these lines are perpendicular
You cannot tell without a graph.
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.
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?
Choices: Pay per visit, Buy a membership, 2, 5, 7, 11
1) For 3 visits:
- Pay per visit: $28 x 3 = $84
- Buying a membership and paying $15 per visit: $90 + ($15 x 3) = $90 + $45 = $135
In this case, it is best to pay per visit as it is cheaper ($84 < $135).
2) For 8 visits:
- Pay per visit: $28 x 8 = $224
- Buying a membership and paying $15 per visit: $90 + ($15 x 8) = $90 + $120 = $210
In this case, it is best to buy a membership and pay per visit as it is cheaper ($210 < $224).
3) With a budget of $150:
- Pay per visit: $28 x X ≤ $150 (where X represents the number of visits)
Solving for X: X ≤ $150 / $28 = approximately 5.36
Since you cannot have a fraction of a visit, the best option would be to buy a membership and pay per visit.
4) To find the break-even point where both options cost about the same:
- Pay per visit: $28 x X
- Buying a membership and paying $15 per visit: $90 + ($15 x X)
Setting the two options equal to each other and solving for X:
$28X = $90 + ($15X)
$28X - $15X = $90
$13X = $90
X = $90 / $13
X ≈ 6.92
Approximately 7 visits would be the break-even point where both options would cost about the same.
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 of solving a system of linear equations.
There are three commonly used methods: graphing, substitution, and elimination.
Some people find graphing to be an easier method as it provides a visual representation of the equations, allowing them to easily identify the point of intersection (or lack thereof) between the two lines. However, graphing can be time-consuming and less accurate when dealing with complex equations or decimal solutions.
Substitution involves solving one equation for one variable and substituting it into the other equation. This method can be useful when one equation is already solved for a variable, making it simple to substitute and find the solution step-by-step.
Elimination is another popular method where the goal is to eliminate one variable by adding or subtracting the equations. This can simplify the system and allow for solving the remaining variable easily. It is often preferred when the coefficients of the variables in the equations easily cancel each other out.
Each method has its advantages and disadvantages, and the choice of method may depend on the specific problem and personal preference. Some people may find substitution easier, while others may prefer elimination based on their own understanding and comfort with the techniques involved.
To determine if the line through (3, 7) and (-2, -5) intersects with the line through (4, 8) and (10, -2), we can use the slope-intercept form of a line.
First, let's find the slope of the first line:
Slope of the first line = (change in y) / (change in x)
= (-5 - 7) / (-2 - 3)
= -12 / -5
= 12/5
Now, let's find the slope of the second line:
Slope of the second line = (change in y) / (change in x)
= (-2 - 8) / (10 - 4)
= -10 / 6
= -5/3
Since the slopes of these two lines, 12/5 and -5/3, are not equal, the lines are not parallel. Therefore, the correct answer is:
Yes…these lines are not parallel.
To determine whether the lines intersect or not, we need to compare the slopes of the two lines.
First, let's find the slope of the line passing through (3, 7) and (-2, -5). The slope (m1) can be calculated using the formula:
m1 = (y2 - y1) / (x2 - x1)
Substituting the coordinates, we have:
m1 = (-5 - 7) / (-2 - 3)
= (-12) / (-5)
= 12/5
Now, let's find the slope of the line passing through (4, 8) and (10, -2). The slope (m2) can again be calculated using the same formula:
m2 = (y2 - y1) / (x2 - x1)
Substituting the coordinates, we have:
m2 = (-2 - 8) / (10 - 4)
= (-10) / (6)
= -5/3
Now, since the lines are perpendicular if and only if the product of their slopes is -1, let's check if m1*m2 equals -1:
(12/5) * (-5/3) = -60/15 = -4
Since the product of the slopes is not -1, the lines are not perpendicular.
Next, let's compare the slopes of the two lines to determine if they are parallel. If the slopes are equal, then the lines are parallel.
Comparing m1 and m2:
m1 = 12/5
m2 = -5/3
Since the slopes are not equal, the lines are not parallel.
Therefore, the correct answer is:
No... these lines are not parallel.
Note: To determine if the lines intersect, you will need to plot the points on a graph or perform additional calculations.