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weeks weight

0 1
1 5
2 9
3 13

What does the y-intercept mean in the scenario?

11 answers

In the given scenario, the data appears to represent a linear relationship between the number of weeks and weight. The weight increases as the number of weeks increases.

The y-intercept in this context (the value of weight when weeks = 0) is 1. This means that at the starting point (week 0), the weight is 1 unit.

In real-world terms, the y-intercept can represent the starting weight before any time has passed (i.e., before any weeks have gone by). It provides a baseline or starting value for the weight measurement in this scenario.

b) What does the slope represent in the situation?

In this scenario, the slope represents the rate of change of weight with respect to the number of weeks. To determine the slope, we can look at how much the weight increases for each additional week.

From the provided data points:

  • From week 0 to week 1, the weight increases from 1 to 5 (an increase of 4).
  • From week 1 to week 2, the weight increases from 5 to 9 (an increase of 4).
  • From week 2 to week 3, the weight increases from 9 to 13 (an increase of 4).

This consistent increase of 4 units in weight for each additional week indicates that the slope of the line (change in weight per change in weeks) is 4.

Thus, the slope represents that for every additional week, the weight increases by 4 units. In simpler terms, it quantifies the rate at which weight is gaining over time in this scenario.

What is the solution to this system of linear equations?
y = 3x - 2
y = -x-6
(1 point)
Responses

(-6, 0)
(-6, 0)

(0, -2)
(0, -2)

Infinite solutions
Infinite solutions

No solution
No solution

(-1, -5)
(-1, -5)
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To find the solution to the system of linear equations, we need to set them equal to each other or substitute one into the other. The equations are:

  1. \( y = 3x - 2 \)
  2. \( y = -x - 6 \)

Since both expressions are equal to \( y \), we can set them equal to each other:

\[ 3x - 2 = -x - 6 \]

Now, let's solve for \( x \):

  1. Add \( x \) to both sides: \[ 3x + x - 2 = -6 \] \[ 4x - 2 = -6 \]

  2. Add 2 to both sides: \[ 4x = -4 \]

  3. Divide by 4: \[ x = -1 \]

Now that we have \( x = -1 \), we can substitute this value back into one of the original equations to find \( y \). We'll use the first equation:

\[ y = 3(-1) - 2 \] \[ y = -3 - 2 \] \[ y = -5 \]

Therefore, the solution to the system of equations is \( (-1, -5) \).

Among the given options, the correct response is:

(-1, -5).

A movie theater charges different prices for children and adults. On Friday, 10 adults and 25 children went to the movie and it cost $670. On Sunday 5 adults and 10 children went and it cost $290. (4 points)
a) What equation would represent the cost of going to a movie on Friday?
responses
10a + 25c = 670
5a + 10c = 290
b) What equation would represent the cost of going to a movie on Sunday?
responses
5a + 10c = 290
10a + 25c = 670
c) What would be the best algebraic method to solve given how the equations are set up?
responses
substitution because one of the equations already has a variable solved for
elimination because the equations are already lined up nicely - you could just multiply the Sunday equation by -2 to eliminate the 'a' variable
d) What is the cost of an adult ticket?
responses
58
29
18
22

Let's break down the problem systematically to answer each part.

a) What equation would represent the cost of going to a movie on Friday?

On Friday, we have 10 adults and 25 children, with a total cost of $670. We can express this as:

\[ 10a + 25c = 670 \]

Response: 10a + 25c = 670

b) What equation would represent the cost of going to a movie on Sunday?

On Sunday, we have 5 adults and 10 children, with a total cost of $290. We can express this as:

\[ 5a + 10c = 290 \]

Response: 5a + 10c = 290

c) What would be the best algebraic method to solve given how the equations are set up?

The equations are structured nicely for elimination, especially since both equations can easily be manipulated to eliminate one of the variables. You can multiply the Sunday equation by 2, to line them up and eliminate the 'c' variable or manipulate them further to eliminate 'a'.

Response: elimination because the equations are already lined up nicely - you could just multiply the Sunday equation by -2 to eliminate the 'a' variable

d) What is the cost of an adult ticket?

Now, let’s solve the system of equations.

We have the following two equations:

  1. \( 10a + 25c = 670 \) (1)
  2. \( 5a + 10c = 290 \) (2)

To eliminate \( c \), we can first simplify equation (2):

Dividing equation (2) by 5:

\[ a + 2c = 58 \quad (3) \]

Now we can multiply equation (3) by -10 to align with equation (1):

\[ -10a - 20c = -580 \quad (4) \]

Now, we can add equation (1) and equation (4):

\[ (10a + 25c) + (-10a - 20c) = 670 - 580 \]

This reduces to:

\[ 5c = 90 \]

Dividing by 5 gives:

\[ c = 18 \]

Now that we know \( c \), plug it back into equation (3):

\[ a + 2(18) = 58 \] \[ a + 36 = 58 \] \[ a = 22 \]

Therefore, the cost of an adult ticket is 22.

Response: 22

y = x + 20
a) How would you describe the relationship between the variables shown?

The equation \( y = x + 20 \) describes a linear relationship between the variables \( x \) and \( y \). Here's a breakdown of the relationship:

  1. Linear Relationship: The equation represents a straight line when graphed on a coordinate plane. This means that the change in \( y \) is consistent as \( x \) changes, which is characteristic of a linear function.

  2. Slope: The slope of the line is \( 1 \), which indicates that for every increase of \( 1 \) in \( x \), \( y \) increases by \( 1 \) as well.

  3. Y-Intercept: The constant term \( +20 \) indicates that the line intersects the y-axis at the point (0, 20). This means that when \( x = 0 \), \( y \) will be \( 20 \).

  4. Positive Correlation: Since the slope is positive, there is a positive correlation between \( x \) and \( y \). As \( x \) increases, \( y \) also increases.

In summary, the relationship between the variables \( x \) and \( y \) is a positive linear relationship with a slope of 1 and a y-intercept of 20.

is y=2x3 a function or not a function?

The expression \( y = 2x^3 \) describes a relationship between the variables \( x \) and \( y \). To determine whether it represents a function, we can use the definition of a function.

A relation is a function if for each input (or value of \( x \)), there is exactly one output (or value of \( y \)).

Analysis of \( y = 2x^3 \):

  1. Input Values: For every real number value of \( x \), you can calculate exactly one corresponding value of \( y \) by plugging the value of \( x \) into the equation.

  2. Graph: If you were to graph the equation \( y = 2x^3 \), it would produce a curve that passes through the origin (0, 0) and continues infinitely in the positive and negative directions along both axes. Each vertical line drawn through the graph would intersect it at exactly one point. This further supports that each \( x \) has exactly one \( y \).

Since for every \( x \) in the domain (the set of all real numbers \( x \)), there is a unique corresponding \( y \) value, we conclude that:

Conclusion:

Yes, \( y = 2x^3 \) is a function.