Question
A tiling company completes two jobs. The first job has $1200 in labor expenses for 40 hours worked, while the second job has $1560 in labor expenses for 52 hours worked. The relationship between the labor expenses and the hours worked is linear. Which equation can be used to calculate the y-intercept of the linear equation?
1200 = 40 (40) + b
1560 = 30 (40) + b
1560 = 30 (52) + b
1200 = 52 (30) + b
1200 = 40 (40) + b
1560 = 30 (40) + b
1560 = 30 (52) + b
1200 = 52 (30) + b
Answers
GPT-4o mini
To find the equation that can be used to calculate the y-intercept of the linear relationship between labor expenses (y) and hours worked (x), we'll first need to determine the slope (m) of the line that connects the two points corresponding to the jobs.
The two points from the jobs can be represented as:
- Job 1: (40 hours, $1200) → point 1: (40, 1200)
- Job 2: (52 hours, $1560) → point 2: (52, 1560)
The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the values from the two jobs:
\[
m = \frac{1560 - 1200}{52 - 40} = \frac{360}{12} = 30
\]
Now that we have the slope (m = 30), we can use the point-slope form of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the first job point (40, 1200), we get:
\[
y - 1200 = 30(x - 40)
\]
Now we can convert this to slope-intercept form (y = mx + b):
1. Distribute the 30:
\[
y - 1200 = 30x - 1200
\]
2. Add 1200 to both sides:
\[
y = 30x
\]
From here, we can see that the y-intercept (b) is 0. However, we want to express the y-intercept calculation based on the existing equations given in the problem.
Given the options presented, let's analyze them focusing on deriving the equation for labor expenses as a function of hours (using the slope we calculated):
1. \( 1200 = 40(30) + b \)
2. \( 1560 = 30(40) + b \)
3. \( 1560 = 30(52) + b \)
4. \( 1200 = 52(30) + b \)
Since we computed that the slope (m) is 30, and we know the formula:
\[
y = mx + b \implies b = y - mx
\]
We should be able to effectively rearrange one of the equations to solve for b.
The first equation correctly calculates b using Job 1:
\[
1200 = 30(40) + b \implies 1200 = 1200 + b \implies b = 0
\]
Thus, the equation that legitimately helps in calculating the y-intercept (b) is:
\[
1200 = 30(40) + b
\]
This confirms that the first equation can be used to find \( b \) effectively, leading us to conclude the equation we need is:
\[
\boxed{1200 = 30(40) + b}
\]
The two points from the jobs can be represented as:
- Job 1: (40 hours, $1200) → point 1: (40, 1200)
- Job 2: (52 hours, $1560) → point 2: (52, 1560)
The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the values from the two jobs:
\[
m = \frac{1560 - 1200}{52 - 40} = \frac{360}{12} = 30
\]
Now that we have the slope (m = 30), we can use the point-slope form of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the first job point (40, 1200), we get:
\[
y - 1200 = 30(x - 40)
\]
Now we can convert this to slope-intercept form (y = mx + b):
1. Distribute the 30:
\[
y - 1200 = 30x - 1200
\]
2. Add 1200 to both sides:
\[
y = 30x
\]
From here, we can see that the y-intercept (b) is 0. However, we want to express the y-intercept calculation based on the existing equations given in the problem.
Given the options presented, let's analyze them focusing on deriving the equation for labor expenses as a function of hours (using the slope we calculated):
1. \( 1200 = 40(30) + b \)
2. \( 1560 = 30(40) + b \)
3. \( 1560 = 30(52) + b \)
4. \( 1200 = 52(30) + b \)
Since we computed that the slope (m) is 30, and we know the formula:
\[
y = mx + b \implies b = y - mx
\]
We should be able to effectively rearrange one of the equations to solve for b.
The first equation correctly calculates b using Job 1:
\[
1200 = 30(40) + b \implies 1200 = 1200 + b \implies b = 0
\]
Thus, the equation that legitimately helps in calculating the y-intercept (b) is:
\[
1200 = 30(40) + b
\]
This confirms that the first equation can be used to find \( b \) effectively, leading us to conclude the equation we need is:
\[
\boxed{1200 = 30(40) + b}
\]