You can make 5 gal of liquid fertilizer by mixing 8 tsp of powdered fertilizer with water. Represent the relation between the teaspoons of powder used and the gallons of fertilizer made using a table, an equation, and a graph is the amount of fertilizer made a function of the amount of powder used?
Number of tsp, x Number of gal,y
0 0
8 5
16 10
24 15
Represent the relation using an equation. y = __ (Simplify your answer. Type an expression using x as the variable. Use integers or fractions for any numbers in the expression.)
To represent the relation using an equation, we can assume that the relationship is linear.
From the table, we can see that when the number of tsp is 0, the number of gal is also 0. This gives us the point (0,0).
Using the point (8,5), we can find the slope of the line:
slope = (change in y) / (change in x) = (5-0) / (8-0) = 5/8
Using the slope-intercept form of a linear equation, y = mx + b, we can plug in the slope and the coordinates of one point (0,0) to find the y-intercept (b).
0 = (5/8)(0) + b
0 = 0 + b
b = 0
So the equation representing the relation between the teaspoons of powder used and the gallons of fertilizer made is: y = (5/8)x
To represent the relation between the teaspoons of powder used (x) and the gallons of fertilizer made (y) using an equation, we can observe that the relationship is linear. This is because as the amount of powder used increases, the amount of fertilizer made also increases, maintaining a constant ratio.
To find the constant ratio, we can compare the values in the table. We notice that for every 8 teaspoons of powder used, 5 gallons of fertilizer are made. Therefore, the ratio of teaspoons to gallons is 8/5 or 8:5.
We can express this relationship in the form of a linear equation using the slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept.
Using the given information, we can determine that the slope of the linear equation is 8/5. Therefore, our equation becomes:
y = (8/5)x + b
To find the value of the y-intercept (b), we can substitute one of the data points from the table. Let's use the first data point (0,0):
0 = (8/5)(0) + b
0 = 0 + b
b = 0
We conclude that the value of the y-intercept (b) is 0. Therefore, the equation representing the relationship between the teaspoons of powder used (x) and the gallons of fertilizer made (y) is:
y = (8/5)x
To represent the relation between the teaspoons of powder used, x, and the gallons of fertilizer made, y, you can use the data points provided to create a table, an equation, and a graph.
Table:
Number of tsp, x | Number of gal,y
0 | 0
8 | 5
16 | 10
24 | 15
Equation:
To find the equation, we need to determine the relationship between x and y. From the given data, we can see that for every 8 tsp of powdered fertilizer, 5 gallons of liquid fertilizer are produced. This indicates a linear relationship with a constant rate of change.
The rate of change, or slope, can be calculated by taking the difference in the gallons of fertilizer (y) and dividing it by the difference in teaspoons of powder (x). Using any two points from the table, let's take the first and second points (0,0) and (8,5):
Slope = (change in y)/(change in x) = (5-0)/(8-0) = 5/8
Now, we have the slope of the line. To find the equation of the line, we use the point-slope form y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Let's use the first point (0,0):
y - 0 = (5/8)(x - 0)
y = (5/8)x
Therefore, the equation that represents the relation between the teaspoons of powder used, x, and the gallons of fertilizer made, y, is y = (5/8)x.
Graph:
To graph this equation, plot the points from the table and draw a line through them. The x-axis represents the number of teaspoons of powder used, and the y-axis represents the number of gallons of fertilizer made.
The graph should show a straight line passing through the points (0,0), (8,5), (16,10), and (24,15), indicating a linear relationship between x and y.