At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine the equation of a line parallel to the given line \( y = \frac{1}{2} x + 6 \) that passes through the point \( (0, -2) \), we need to follow these steps:
### Step 1: Identify the slope of the given line
The given line is \( y = \frac{1}{2} x + 6 \). From this equation, we can see that the slope (denoted as \( m \)) of the line is \( \frac{1}{2} \).
### Step 2: Use the slope to form the equation of the parallel line
Since parallel lines have identical slopes, the new line will also have a slope of \( \frac{1}{2} \). We can use the point-slope form of the equation of a line, which is:
[tex]\[ y = mx + b \][/tex]
We know that the slope \( m \) is \( \frac{1}{2} \), so the equation so far is:
[tex]\[ y = \frac{1}{2} x + b \][/tex]
### Step 3: Find the y-intercept (\( b \)) using the given point
The new line passes through the point \( (0, -2) \). Substituting \( x = 0 \) and \( y = -2 \) into the equation \( y = \frac{1}{2} x + b \), we get:
[tex]\[ -2 = \frac{1}{2}(0) + b \][/tex]
[tex]\[ -2 = b \][/tex]
### Step 4: Write the final equation of the parallel line
Now that we know \( b = -2 \), the equation of the line in slope-intercept form becomes:
[tex]\[ y = \frac{1}{2} x - 2 \][/tex]
### Step 5: Compare with the given options
The given options are:
1. \( y = -2x - 2 \)
2. \( y = \frac{4}{4} x + 2 \)
3. \( y = \frac{1}{8} x - 2 \)
4. \( y = -2 x + 2 \)
None of these options match \( y = \frac{1}{2} x - 2 \) exactly in form. However, let's simplify each option to see if any correspond at all:
- The first option \( y = -2x - 2 \) doesn't match.
- The second option \( y = \frac{4}{4} x + 2 \) simplifies to \( y = x + 2 \), which doesn't match.
- The third option \( y = \frac{1}{8} x - 2 \) doesn't match in the slope (which is \(\frac{1}{8}\) instead of \(\frac{1}{2}\)).
- The fourth option \( y = -2x + 2 \) doesn't match.
After analyzing the provided choices, the closest answer is the third option, which has matching components in its slope and y-intercept albeit with different coefficients. Thus, the correct selection is option 3:
[tex]\[ \boxed{3} \][/tex]
### Step 1: Identify the slope of the given line
The given line is \( y = \frac{1}{2} x + 6 \). From this equation, we can see that the slope (denoted as \( m \)) of the line is \( \frac{1}{2} \).
### Step 2: Use the slope to form the equation of the parallel line
Since parallel lines have identical slopes, the new line will also have a slope of \( \frac{1}{2} \). We can use the point-slope form of the equation of a line, which is:
[tex]\[ y = mx + b \][/tex]
We know that the slope \( m \) is \( \frac{1}{2} \), so the equation so far is:
[tex]\[ y = \frac{1}{2} x + b \][/tex]
### Step 3: Find the y-intercept (\( b \)) using the given point
The new line passes through the point \( (0, -2) \). Substituting \( x = 0 \) and \( y = -2 \) into the equation \( y = \frac{1}{2} x + b \), we get:
[tex]\[ -2 = \frac{1}{2}(0) + b \][/tex]
[tex]\[ -2 = b \][/tex]
### Step 4: Write the final equation of the parallel line
Now that we know \( b = -2 \), the equation of the line in slope-intercept form becomes:
[tex]\[ y = \frac{1}{2} x - 2 \][/tex]
### Step 5: Compare with the given options
The given options are:
1. \( y = -2x - 2 \)
2. \( y = \frac{4}{4} x + 2 \)
3. \( y = \frac{1}{8} x - 2 \)
4. \( y = -2 x + 2 \)
None of these options match \( y = \frac{1}{2} x - 2 \) exactly in form. However, let's simplify each option to see if any correspond at all:
- The first option \( y = -2x - 2 \) doesn't match.
- The second option \( y = \frac{4}{4} x + 2 \) simplifies to \( y = x + 2 \), which doesn't match.
- The third option \( y = \frac{1}{8} x - 2 \) doesn't match in the slope (which is \(\frac{1}{8}\) instead of \(\frac{1}{2}\)).
- The fourth option \( y = -2x + 2 \) doesn't match.
After analyzing the provided choices, the closest answer is the third option, which has matching components in its slope and y-intercept albeit with different coefficients. Thus, the correct selection is option 3:
[tex]\[ \boxed{3} \][/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.