Thinking With Math Models Unit Test

Explanation
The equation of a line can be found using the formula y = mx + b, where m is the slope of the line and b is the y-intercept. To find the slope, we can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the given points. In this case, the slope is (4 - 3) / (2 - 1) = 1. Since the line passes through the point (1, 3), we can substitute these values into the equation to find the y-intercept. 3 = 1(1) + b, which gives us b = 2. Therefore, the equation of the line is y = x + 2.

Explanation
The equation of a line can be found using the formula y = mx + b, where m is the slope of the line and b is the y-intercept. To find the slope, we can use the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates are (0, 5) and (1, 8). Substituting these values into the formula, we get (8 - 5) / (1 - 0) = 3/1 = 3. Therefore, the slope of the line is 3. Substituting the slope and the coordinates of one point (0, 5) into the equation y = mx + b, we can solve for b. 5 = 3(0) + b, which gives us b = 5. Therefore, the equation of the line is y = 3x + 5.

Explanation
Use the equation y=mx+b

Plug slope in for m

Use the x and y values from the coordinate point to plug into y=mx+b

Solve for b

Explanation
The equation y=10/x represents the relationship between the number of cups (y) and the price per package (x). This equation states that the value of y is equal to 10 divided by x. This means that as the price per package increases (x increases), the number of cups decreases (y decreases), and vice versa. The equation y=10/x accurately describes this inverse relationship between the number of cups and the price per package.

Explanation
It is inverse because the x value times the y value gives you the constant of 30 each time

Explanation
It is direct because if you take the y value divided by the x value, you get the constant of 7 each time.

Explanation
The relationship represented by the given table is linear. This is because the values in the table show a consistent pattern of change, with a constant rate of increase or decrease. In a linear relationship, there is a direct correlation between the input and output values, and the graph of the relationship would be a straight line.

Explanation
The equation c=100h+200 represents the situation where c is the total cost of renting the photo booth, h is the number of hours the booth is rented for, and 200 is the fixed cost for the booth to come to the location. The additional cost of $100 per hour is represented by 100h, as it is multiplied by the number of hours h. Adding the fixed cost of $200 to the additional cost gives the total cost of renting the photo booth.

Explanation
The equation y=30/x models the data above because it represents a relationship where the value of y is inversely proportional to the value of x. As x increases, y decreases and vice versa. This is evident in the given data where as x increases, y decreases. The equation y=30/x captures this relationship accurately.

Explanation
The equation y=5/2x-2 models the given data because it represents a linear relationship between x and y. The slope of the line is 5/2, which means that for every increase of 2 in x, y increases by 5. The y-intercept is -2, which means that when x is 0, y is -2. This equation accurately represents the pattern observed in the data.

Explanation
The equation y=7x models the given data because it represents a linear relationship between the variables x and y, where y is equal to 7 times x. This means that as x increases, y also increases at a constant rate. The equation y=7/x does not model the data because it represents a reciprocal relationship, where y decreases as x increases. The equation y=10/x does not model the data because it represents a reciprocal relationship with a different constant value. The equation y=7x+1 does not model the data because it includes an additional constant term of 1.