Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine the equation of the line that is perpendicular to [tex]\( y = 4x + 6 \)[/tex] and passes through the point [tex]\( (8, -4) \)[/tex], follow these steps:
1. Find the slope of the given line:
The given line equation is [tex]\( y = 4x + 6 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Therefore, the slope of the given line is [tex]\( m = 4 \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the given slope. Thus, the slope of the line perpendicular to the given line is [tex]\( m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{4} \)[/tex].
3. Use the point-slope form to write the equation of the perpendicular line:
The point-slope form of a line's equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the point the line passes through, and [tex]\( m \)[/tex] is the slope of the line. In this case, the point is [tex]\( (8, -4) \)[/tex] and the slope is [tex]\( -\frac{1}{4} \)[/tex].
4. Substitute the given point and the slope into the point-slope form:
[tex]\[ y - (-4) = -\frac{1}{4}(x - 8) \][/tex]
Simplifying this equation step-by-step:
[tex]\[ y + 4 = -\frac{1}{4}(x - 8) \][/tex]
Distribute the [tex]\( -\frac{1}{4} \)[/tex]:
[tex]\[ y + 4 = -\frac{1}{4}x + 2 \][/tex]
Subtract 4 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{4}x + 2 - 4 \][/tex]
Simplify the expression:
[tex]\[ y = -\frac{1}{4}x - 2 \][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y = 4x + 6 \)[/tex] and passes through the point [tex]\( (8, -4) \)[/tex] is [tex]\( y = -\frac{1}{4}x - 2 \)[/tex].
The correct choice is:
[tex]\[ \boxed{y = -\frac{1}{4}x - 2} \][/tex]
1. Find the slope of the given line:
The given line equation is [tex]\( y = 4x + 6 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Therefore, the slope of the given line is [tex]\( m = 4 \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the given slope. Thus, the slope of the line perpendicular to the given line is [tex]\( m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{4} \)[/tex].
3. Use the point-slope form to write the equation of the perpendicular line:
The point-slope form of a line's equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the point the line passes through, and [tex]\( m \)[/tex] is the slope of the line. In this case, the point is [tex]\( (8, -4) \)[/tex] and the slope is [tex]\( -\frac{1}{4} \)[/tex].
4. Substitute the given point and the slope into the point-slope form:
[tex]\[ y - (-4) = -\frac{1}{4}(x - 8) \][/tex]
Simplifying this equation step-by-step:
[tex]\[ y + 4 = -\frac{1}{4}(x - 8) \][/tex]
Distribute the [tex]\( -\frac{1}{4} \)[/tex]:
[tex]\[ y + 4 = -\frac{1}{4}x + 2 \][/tex]
Subtract 4 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{4}x + 2 - 4 \][/tex]
Simplify the expression:
[tex]\[ y = -\frac{1}{4}x - 2 \][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y = 4x + 6 \)[/tex] and passes through the point [tex]\( (8, -4) \)[/tex] is [tex]\( y = -\frac{1}{4}x - 2 \)[/tex].
The correct choice is:
[tex]\[ \boxed{y = -\frac{1}{4}x - 2} \][/tex]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.