Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine which equation represents a line that is parallel to [tex]\( y = \frac{1}{2} x - 2 \)[/tex] and passes through the point [tex]\((-8, 1)\)[/tex], we need to follow these steps:
### Step 1: Identify the Slope of the Given Line
The equation [tex]\( y = \frac{1}{2} x - 2 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. From this equation, we can see that the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
### Step 2: Understand the Condition for Parallel Lines
Lines that are parallel have the same slope. Therefore, the slope of our new line must also be [tex]\( \frac{1}{2} \)[/tex].
### Step 3: Apply the Point-Slope Form
The line passes through the point [tex]\((-8, 1)\)[/tex], so we can use the point-slope form of the equation of a line:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is [tex]\((-8, 1)\)[/tex] and [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Substitute the values into the point-slope form:
[tex]\[ y - 1 = \frac{1}{2}(x - (-8)) \][/tex]
[tex]\[ y - 1 = \frac{1}{2}(x + 8) \][/tex]
### Step 4: Simplify to Slope-Intercept Form
Solve for [tex]\( y \)[/tex] to convert this equation to slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[ y - 1 = \frac{1}{2}x + \frac{1}{2} \times 8 \][/tex]
[tex]\[ y - 1 = \frac{1}{2}x + 4 \][/tex]
[tex]\[ y = \frac{1}{2}x + 4 + 1 \][/tex]
[tex]\[ y = \frac{1}{2}x + 5 \][/tex]
### Step 5: Identify the Correct Option
The equation we derived is [tex]\( y = \frac{1}{2} x + 5 \)[/tex]. Now, we check the given options:
[tex]\[ \begin{array}{c} 1. \ y = \frac{1}{2} x + 5 \\ 2. \ y = \frac{1}{2} x - 9 \\ 3. \ y = -2 x - 7 \\ 4. \ y = -\frac{1}{2} x + 5 \\ 5. \ y = -2 x - 5 \\ \end{array} \][/tex]
The correct equation is:
[tex]\[ y = \frac{1}{2} x + 5 \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{1} \][/tex]
### Step 1: Identify the Slope of the Given Line
The equation [tex]\( y = \frac{1}{2} x - 2 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. From this equation, we can see that the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
### Step 2: Understand the Condition for Parallel Lines
Lines that are parallel have the same slope. Therefore, the slope of our new line must also be [tex]\( \frac{1}{2} \)[/tex].
### Step 3: Apply the Point-Slope Form
The line passes through the point [tex]\((-8, 1)\)[/tex], so we can use the point-slope form of the equation of a line:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is [tex]\((-8, 1)\)[/tex] and [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Substitute the values into the point-slope form:
[tex]\[ y - 1 = \frac{1}{2}(x - (-8)) \][/tex]
[tex]\[ y - 1 = \frac{1}{2}(x + 8) \][/tex]
### Step 4: Simplify to Slope-Intercept Form
Solve for [tex]\( y \)[/tex] to convert this equation to slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[ y - 1 = \frac{1}{2}x + \frac{1}{2} \times 8 \][/tex]
[tex]\[ y - 1 = \frac{1}{2}x + 4 \][/tex]
[tex]\[ y = \frac{1}{2}x + 4 + 1 \][/tex]
[tex]\[ y = \frac{1}{2}x + 5 \][/tex]
### Step 5: Identify the Correct Option
The equation we derived is [tex]\( y = \frac{1}{2} x + 5 \)[/tex]. Now, we check the given options:
[tex]\[ \begin{array}{c} 1. \ y = \frac{1}{2} x + 5 \\ 2. \ y = \frac{1}{2} x - 9 \\ 3. \ y = -2 x - 7 \\ 4. \ y = -\frac{1}{2} x + 5 \\ 5. \ y = -2 x - 5 \\ \end{array} \][/tex]
The correct equation is:
[tex]\[ y = \frac{1}{2} x + 5 \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{1} \][/tex]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.