1. Gina wants to begin by figuring out how long the corn bed will be. She knows that in her previous two gardens, she was able to plant 2 rows in a 5 foot long space and 5 rows in an 11 foot long space. Find a linear equation for the length of the bed (y) in which she can fit x rows of corn.
Hint: Use the two pairs of values from her previous two gardens to write two ordered pairs of the form (x,y), then use these pairs to write an equation of the form y = mx + b.
To find a linear equation for the length of the bed (y) in terms of the number of rows of corn (x), we can use the two pairs of values provided from Gina's previous gardens.
Let's write down the two pairs of values as ordered pairs in the form (x, y):
First garden: 2 rows in a 5-foot long space => (2, 5)
Second garden: 5 rows in an 11-foot long space => (5, 11)
Now, let's use these two points to determine the equation of the line.
Step 1: Find the slope (m)
The slope of a line is calculated using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line.
Substituting the values from the first and second gardens, we have:
m = (11 - 5) / (5 - 2)
m = 6 / 3
m = 2
So, the slope of the line is 2.
Step 2: Find the y-intercept (b)
The y-intercept is the value of y when x = 0. To find it, we can substitute the coordinates of one of the points in the equation y = mx + b and solve for b.
Using the point (2, 5):
5 = 2 * 2 + b
5 = 4 + b
b = 5 - 4
b = 1
So, the y-intercept (b) is 1.
Step 3: Write the equation of the line
Now that we have the slope (m = 2) and the y-intercept (b = 1), we can put it all together to form the linear equation in the form y = mx + b:
y = 2x + 1
Therefore, the linear equation for the length of the corn bed (y) in terms of the number of rows of corn (x) is y = 2x + 1.