Answer :
Certainly! Let's solve the problem step-by-step to find the equation of the line passing through the point (9, 5) and parallel to the line given by the equation [tex]\(6x - y = 5\)[/tex].
### Step-by-Step Solution
#### 1. Determine the slope of the given line [tex]\(6x - y = 5\)[/tex].
The given equation of the line is in the form [tex]\(Ax + By = C\)[/tex]. The slope [tex]\(m\)[/tex] of a line [tex]\(Ax + By = C\)[/tex] can be found using the formula:
[tex]\[ m = -\frac{A}{B} \][/tex]
For the equation [tex]\(6x - y = 5\)[/tex],
[tex]\[ A = 6 \][/tex]
[tex]\[ B = -1 \][/tex]
So, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = -\frac{6}{-1} = 6 \][/tex]
#### 2. Use the point-slope form of the line to find the equation passing through the given point (9, 5).
The point-slope form of an equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope.
Here, [tex]\((x_1, y_1) = (9, 5)\)[/tex], and [tex]\(m = 6\)[/tex]. Plugging in these values, we get:
[tex]\[ y - 5 = 6(x - 9) \][/tex]
#### 3. Simplify the equation to the slope-intercept form [tex]\(y = mx + b\)[/tex].
First, expand and simplify the equation:
[tex]\[ y - 5 = 6x - 54 \][/tex]
[tex]\[ y = 6x - 54 + 5 \][/tex]
[tex]\[ y = 6x - 49 \][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[ y = 6x - 49 \][/tex]
#### 4. Convert the equation to standard form [tex]\(Ax + By = C\)[/tex].
We start from the slope-intercept form [tex]\( y = 6x - 49 \)[/tex] and rearrange it to the form [tex]\(Ax + By = C\)[/tex]:
[tex]\[ y = 6x - 49 \][/tex]
Move all terms to one side to get:
[tex]\[ 6x - y = 49 \][/tex]
So, the equation in standard form is:
[tex]\[ 6x - y = 49 \][/tex]
### Summary
(a) In slope-intercept form: [tex]\( y = 6x - 49 \)[/tex]
(b) In standard form: [tex]\( 6x - y = 49 \)[/tex]
These are the equations of the line passing through the point (9, 5) and parallel to [tex]\(6x - y = 5\)[/tex].
### Step-by-Step Solution
#### 1. Determine the slope of the given line [tex]\(6x - y = 5\)[/tex].
The given equation of the line is in the form [tex]\(Ax + By = C\)[/tex]. The slope [tex]\(m\)[/tex] of a line [tex]\(Ax + By = C\)[/tex] can be found using the formula:
[tex]\[ m = -\frac{A}{B} \][/tex]
For the equation [tex]\(6x - y = 5\)[/tex],
[tex]\[ A = 6 \][/tex]
[tex]\[ B = -1 \][/tex]
So, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = -\frac{6}{-1} = 6 \][/tex]
#### 2. Use the point-slope form of the line to find the equation passing through the given point (9, 5).
The point-slope form of an equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope.
Here, [tex]\((x_1, y_1) = (9, 5)\)[/tex], and [tex]\(m = 6\)[/tex]. Plugging in these values, we get:
[tex]\[ y - 5 = 6(x - 9) \][/tex]
#### 3. Simplify the equation to the slope-intercept form [tex]\(y = mx + b\)[/tex].
First, expand and simplify the equation:
[tex]\[ y - 5 = 6x - 54 \][/tex]
[tex]\[ y = 6x - 54 + 5 \][/tex]
[tex]\[ y = 6x - 49 \][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[ y = 6x - 49 \][/tex]
#### 4. Convert the equation to standard form [tex]\(Ax + By = C\)[/tex].
We start from the slope-intercept form [tex]\( y = 6x - 49 \)[/tex] and rearrange it to the form [tex]\(Ax + By = C\)[/tex]:
[tex]\[ y = 6x - 49 \][/tex]
Move all terms to one side to get:
[tex]\[ 6x - y = 49 \][/tex]
So, the equation in standard form is:
[tex]\[ 6x - y = 49 \][/tex]
### Summary
(a) In slope-intercept form: [tex]\( y = 6x - 49 \)[/tex]
(b) In standard form: [tex]\( 6x - y = 49 \)[/tex]
These are the equations of the line passing through the point (9, 5) and parallel to [tex]\(6x - y = 5\)[/tex].
