Question
What is the y-intercept of the graphed linear function?
graph y-3=1/2(x+2)
Answers
Answered by
GPT-4o mini
To determine the y-intercept of the linear function given by the equation \(y - 3 = \frac{1}{2}(x + 2)\), we first need to rewrite it in slope-intercept form, which is \(y = mx + b\), where \(b\) is the y-intercept.
1. Start with the original equation:
\[
y - 3 = \frac{1}{2}(x + 2)
\]
2. Distribute \(\frac{1}{2}\) on the right side:
\[
y - 3 = \frac{1}{2}x + 1
\]
3. Add 3 to both sides:
\[
y = \frac{1}{2}x + 1 + 3
\]
\[
y = \frac{1}{2}x + 4
\]
Now that we have the equation in slope-intercept form, we can identify the y-intercept. In this case, the y-intercept \(b\) is 4.
Thus, the y-intercept of the graphed linear function is \(4\).
1. Start with the original equation:
\[
y - 3 = \frac{1}{2}(x + 2)
\]
2. Distribute \(\frac{1}{2}\) on the right side:
\[
y - 3 = \frac{1}{2}x + 1
\]
3. Add 3 to both sides:
\[
y = \frac{1}{2}x + 1 + 3
\]
\[
y = \frac{1}{2}x + 4
\]
Now that we have the equation in slope-intercept form, we can identify the y-intercept. In this case, the y-intercept \(b\) is 4.
Thus, the y-intercept of the graphed linear function is \(4\).
Answer
Use the graph to find the x-intercept for the equation y + 2 = 2(x + 3).
The x-intercept is
.
The x-intercept is
.
Answered by
GPT-4o mini
To find the x-intercept of the equation \(y + 2 = 2(x + 3)\), we need to set \(y = 0\) and solve for \(x\).
1. Start with the equation:
\[
y + 2 = 2(x + 3)
\]
2. Set \(y = 0\):
\[
0 + 2 = 2(x + 3)
\]
\[
2 = 2(x + 3)
\]
3. Divide both sides by 2:
\[
1 = x + 3
\]
4. Solve for \(x\):
\[
x = 1 - 3
\]
\[
x = -2
\]
Thus, the x-intercept of the equation is \(-2\).
1. Start with the equation:
\[
y + 2 = 2(x + 3)
\]
2. Set \(y = 0\):
\[
0 + 2 = 2(x + 3)
\]
\[
2 = 2(x + 3)
\]
3. Divide both sides by 2:
\[
1 = x + 3
\]
4. Solve for \(x\):
\[
x = 1 - 3
\]
\[
x = -2
\]
Thus, the x-intercept of the equation is \(-2\).
Answer
what is for y?
Answered by
GPT-4o mini
It seems like you would like to find the y-intercept for the given equation \( y + 2 = 2(x + 3) \).
To find the y-intercept, we set \( x = 0 \) in the equation and solve for \( y \):
1. Start with the original equation:
\[
y + 2 = 2(x + 3)
\]
2. Set \( x = 0 \):
\[
y + 2 = 2(0 + 3)
\]
\[
y + 2 = 2 \cdot 3
\]
\[
y + 2 = 6
\]
3. Subtract 2 from both sides:
\[
y = 6 - 2
\]
\[
y = 4
\]
So, the y-intercept of the equation is \( 4 \). Thus, the coordinates of the y-intercept are \( (0, 4) \).
To find the y-intercept, we set \( x = 0 \) in the equation and solve for \( y \):
1. Start with the original equation:
\[
y + 2 = 2(x + 3)
\]
2. Set \( x = 0 \):
\[
y + 2 = 2(0 + 3)
\]
\[
y + 2 = 2 \cdot 3
\]
\[
y + 2 = 6
\]
3. Subtract 2 from both sides:
\[
y = 6 - 2
\]
\[
y = 4
\]
So, the y-intercept of the equation is \( 4 \). Thus, the coordinates of the y-intercept are \( (0, 4) \).
Answer
Andy wrote the equation of a line that has a slope of Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction. and passes through the point (3, –2) in function notation.Step 1: y – (–2) = Step 1: y minus left-parenthesis negative 2 right-parenthesis equals StartFraction 3 Over 4 left-parenthesis x minus 3 right-parenthesis. (x – 3)Step 2: y + 2 = Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction.x – Step 3: y + 2 + 2 = Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction.x – + 2Step 4: y = Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction.x – Step 5: f(x) = Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction.x – Analyze each step to identify if Andy made an error. Yes, he made an error in Step 1. He switched the x and y values.Yes, he made an error in Step 2. He did not distribute StartFraction 3 Over 4 EndFraction properly. Yes, he made an error in Step 3. He should have subtracted 2 from both sides. No, his work is correct.
Answered by
GPT-4o mini
Let’s analyze each step of Andy’s work in detail to identify if there are any errors. We'll start with the information given:
- **Slope = \(\frac{3}{4}\)**
- **Point = (3, -2)**
### Step-by-step Analysis
**Step 1: \(y - (-2) = \frac{3}{4}(x - 3)\)**
Here, Andy correctly used the point-slope form of a linear equation, which is given by \(y - y_1 = m(x - x_1)\).
So, substituting in \(m = \frac{3}{4}\), \(x_1 = 3\), and \(y_1 = -2\):
\[
y - (-2) = \frac{3}{4}(x - 3)
\]
This step is correct.
---
**Step 2: \(y + 2 = \frac{3}{4} x - \frac{9}{4}\)**
This step is where Andy appears to make a mistake. He should distribute \(\frac{3}{4}\) to \((x - 3)\).
Applying distribution:
\[
y + 2 = \frac{3}{4} x - \frac{3 \times 3}{4} = \frac{3}{4} x - \frac{9}{4}
\]
He correctly changes \(y - (-2)\) to \(y + 2\) and distributes properly here. So there is no mistake in Step 2.
---
**Step 3: \(y + 2 + 2 = \frac{3}{4} x - \frac{9}{4}\)**
Here, Andy adds 2 to both sides, which seems unnecessary and creates confusion. It should stay as is:
\[
y + 2 = \frac{3}{4} x - \frac{9}{4}
\]
When he says \(y + 2 + 2\), he should instead be working towards isolating \(y\) by subtracting 2. So, this step has an unnecessary addition and potential misunderstanding.
---
**Step 4: \(y = \frac{3}{4} x - \frac{9}{4} - 2\)**
This step is intended to isolate \(y\), but it appears he did not actually perform the subtraction correctly.
To isolate \(y\):
\[
y + 2 = \frac{3}{4} x - \frac{9}{4}
\]
Subtracting 2 from both sides (not shown correctly):
\[
y = \frac{3}{4} x - \frac{9}{4} - 2
\]
To subtract \(2\) from \(-\frac{9}{4}\), convert 2 to quarters:
\[
y = \frac{3}{4} x - \frac{9}{4} - \frac{8}{4}
\]
This gives us:
\[
y = \frac{3}{4} x - \frac{17}{4}
\]
---
**Step 5: \(f(x) = \frac{3}{4} x - \frac{17}{4}\)**
This is the correct form if he had properly worked in Step 4.
### Conclusion
From the analysis:
- **Step 1**: Correct.
- **Step 2**: Correct.
- **Step 3**: Error due to unnecessary addition and misunderstanding of subtraction.
- **Step 4**: Poorly executed isolation of \(y\).
- **Step 5**: Correct based on previously corrected equation.
Overall, **Yes, he made an error in Step 3. He should have subtracted 2 from both sides.** The corrections should lead to a final function representation as \( f(x) = \frac{3}{4} x - \frac{17}{4} \).
- **Slope = \(\frac{3}{4}\)**
- **Point = (3, -2)**
### Step-by-step Analysis
**Step 1: \(y - (-2) = \frac{3}{4}(x - 3)\)**
Here, Andy correctly used the point-slope form of a linear equation, which is given by \(y - y_1 = m(x - x_1)\).
So, substituting in \(m = \frac{3}{4}\), \(x_1 = 3\), and \(y_1 = -2\):
\[
y - (-2) = \frac{3}{4}(x - 3)
\]
This step is correct.
---
**Step 2: \(y + 2 = \frac{3}{4} x - \frac{9}{4}\)**
This step is where Andy appears to make a mistake. He should distribute \(\frac{3}{4}\) to \((x - 3)\).
Applying distribution:
\[
y + 2 = \frac{3}{4} x - \frac{3 \times 3}{4} = \frac{3}{4} x - \frac{9}{4}
\]
He correctly changes \(y - (-2)\) to \(y + 2\) and distributes properly here. So there is no mistake in Step 2.
---
**Step 3: \(y + 2 + 2 = \frac{3}{4} x - \frac{9}{4}\)**
Here, Andy adds 2 to both sides, which seems unnecessary and creates confusion. It should stay as is:
\[
y + 2 = \frac{3}{4} x - \frac{9}{4}
\]
When he says \(y + 2 + 2\), he should instead be working towards isolating \(y\) by subtracting 2. So, this step has an unnecessary addition and potential misunderstanding.
---
**Step 4: \(y = \frac{3}{4} x - \frac{9}{4} - 2\)**
This step is intended to isolate \(y\), but it appears he did not actually perform the subtraction correctly.
To isolate \(y\):
\[
y + 2 = \frac{3}{4} x - \frac{9}{4}
\]
Subtracting 2 from both sides (not shown correctly):
\[
y = \frac{3}{4} x - \frac{9}{4} - 2
\]
To subtract \(2\) from \(-\frac{9}{4}\), convert 2 to quarters:
\[
y = \frac{3}{4} x - \frac{9}{4} - \frac{8}{4}
\]
This gives us:
\[
y = \frac{3}{4} x - \frac{17}{4}
\]
---
**Step 5: \(f(x) = \frac{3}{4} x - \frac{17}{4}\)**
This is the correct form if he had properly worked in Step 4.
### Conclusion
From the analysis:
- **Step 1**: Correct.
- **Step 2**: Correct.
- **Step 3**: Error due to unnecessary addition and misunderstanding of subtraction.
- **Step 4**: Poorly executed isolation of \(y\).
- **Step 5**: Correct based on previously corrected equation.
Overall, **Yes, he made an error in Step 3. He should have subtracted 2 from both sides.** The corrections should lead to a final function representation as \( f(x) = \frac{3}{4} x - \frac{17}{4} \).
Answer
Hugo decided to pay $2 a week to his brother to buy a bicycle. The equation y – 10 = –2(x – 10) models the amount of money he owes for the bike, where x is the number of weeks and y is the amount of money he still needs to pay.
How much did the bicycle cost?
$
After how many weeks will Hugo finish paying for the bike?
weeks
How much did the bicycle cost?
$
After how many weeks will Hugo finish paying for the bike?
weeks
Answered by
GPT-4o mini
To determine the cost of the bicycle and how many weeks it will take for Hugo to finish paying, we can analyze the equation given:
\[
y - 10 = -2(x - 10)
\]
### Step 1: Rewrite the equation in a familiar form
To make sense of this equation, we'll first rewrite it in terms of \(y\):
1. Distribute the -2 on the right-hand side:
\[
y - 10 = -2x + 20
\]
2. Add 10 to both sides:
\[
y = -2x + 20 + 10
\]
\[
y = -2x + 30
\]
### Step 2: Interpreting the equation
In the form \(y = -2x + 30\):
- \(y\) represents the amount of money Hugo still needs to pay for the bicycle.
- The slope \(-2\) indicates that for each week (\(x\)) that passes, the amount he owes decreases by $2.
- The y-intercept (where \(x = 0\)), which is \(30\), represents the total cost of the bicycle.
Thus, **the bicycle cost $30**.
### Step 3: Determine after how many weeks will Hugo finish paying for the bike
Hugo will finish paying for the bicycle when \(y = 0\):
1. Set \(y\) to 0 in the equation:
\[
0 = -2x + 30
\]
2. Solve for \(x\):
\[
2x = 30
\]
\[
x = 15
\]
### Summary of Answers
- The bicycle cost **$30**.
- Hugo will finish paying for the bike after **15 weeks**.
\[
y - 10 = -2(x - 10)
\]
### Step 1: Rewrite the equation in a familiar form
To make sense of this equation, we'll first rewrite it in terms of \(y\):
1. Distribute the -2 on the right-hand side:
\[
y - 10 = -2x + 20
\]
2. Add 10 to both sides:
\[
y = -2x + 20 + 10
\]
\[
y = -2x + 30
\]
### Step 2: Interpreting the equation
In the form \(y = -2x + 30\):
- \(y\) represents the amount of money Hugo still needs to pay for the bicycle.
- The slope \(-2\) indicates that for each week (\(x\)) that passes, the amount he owes decreases by $2.
- The y-intercept (where \(x = 0\)), which is \(30\), represents the total cost of the bicycle.
Thus, **the bicycle cost $30**.
### Step 3: Determine after how many weeks will Hugo finish paying for the bike
Hugo will finish paying for the bicycle when \(y = 0\):
1. Set \(y\) to 0 in the equation:
\[
0 = -2x + 30
\]
2. Solve for \(x\):
\[
2x = 30
\]
\[
x = 15
\]
### Summary of Answers
- The bicycle cost **$30**.
- Hugo will finish paying for the bike after **15 weeks**.
Answer
A tile factory earns money by charging a flat fee for delivery and a sales price of $0.25 per tile. One customer paid a total of $3,000 for 10,000 tiles. The equation y – 3,000 = 0.25(x – 10,000) models the revenue of the tile factory, where x is the number of tiles and y is the total cost to the customer.
Which function describes the revenue of the tile factory in terms of tiles sold?
What is the flat fee for delivery?
$
Which function describes the revenue of the tile factory in terms of tiles sold?
What is the flat fee for delivery?
$
Answered by
GPT-4o mini
To analyze the revenue of the tile factory and identify the flat fee for delivery, we can work with the given equation:
\[
y - 3,000 = 0.25(x - 10,000)
\]
### Step 1: Rewrite the equation
First, we can rewrite the equation into the slope-intercept form \(y = mx + b\):
1. Distribute \(0.25\) on the right side:
\[
y - 3,000 = 0.25x - 0.25 \times 10,000
\]
\[
y - 3,000 = 0.25x - 2,500
\]
2. Add \(3,000\) to both sides:
\[
y = 0.25x - 2,500 + 3,000
\]
\[
y = 0.25x + 500
\]
### Conclusion for Revenue Function
The function that describes the revenue \(y\) of the tile factory in terms of tiles sold \(x\) is:
\[
y = 0.25x + 500
\]
### Step 2: Identify the flat fee for delivery
In the equation \(y = 0.25x + 500\):
- The term \(0.25x\) represents the revenue generated from selling tiles at **$0.25 per tile**.
- The constant term \(500\) represents the flat fee for delivery.
Thus, the flat fee for delivery is **$500**.
### Summary of Answers
- The revenue function in terms of tiles sold is: **\(y = 0.25x + 500\)**
- The flat fee for delivery is: **$500**.
\[
y - 3,000 = 0.25(x - 10,000)
\]
### Step 1: Rewrite the equation
First, we can rewrite the equation into the slope-intercept form \(y = mx + b\):
1. Distribute \(0.25\) on the right side:
\[
y - 3,000 = 0.25x - 0.25 \times 10,000
\]
\[
y - 3,000 = 0.25x - 2,500
\]
2. Add \(3,000\) to both sides:
\[
y = 0.25x - 2,500 + 3,000
\]
\[
y = 0.25x + 500
\]
### Conclusion for Revenue Function
The function that describes the revenue \(y\) of the tile factory in terms of tiles sold \(x\) is:
\[
y = 0.25x + 500
\]
### Step 2: Identify the flat fee for delivery
In the equation \(y = 0.25x + 500\):
- The term \(0.25x\) represents the revenue generated from selling tiles at **$0.25 per tile**.
- The constant term \(500\) represents the flat fee for delivery.
Thus, the flat fee for delivery is **$500**.
### Summary of Answers
- The revenue function in terms of tiles sold is: **\(y = 0.25x + 500\)**
- The flat fee for delivery is: **$500**.