Answer :
To find the equation of a line that is parallel to the given line [tex]\(4x - y = -6\)[/tex] and passes through the point [tex]\((3, 9)\)[/tex], we can follow these steps:
### Step 1: Determine the slope of the given line
First, we will rewrite the equation of the given line [tex]\(4x - y = -6\)[/tex] in slope-intercept form [tex]\(y = mx + b\)[/tex]. This form makes it easier to identify the slope [tex]\(m\)[/tex].
Starting with the equation:
[tex]\[ 4x - y = -6 \][/tex]
Rearrange it to solve for [tex]\(y\)[/tex]:
[tex]\[ -y = -4x - 6 \][/tex]
[tex]\[ y = 4x + 6 \][/tex]
In the slope-intercept form [tex]\(y = mx + b\)[/tex], the coefficient of [tex]\(x\)[/tex] is the slope [tex]\(m\)[/tex]. Here, the slope [tex]\(m\)[/tex] is 4.
### Step 2: Use point-slope form to find the equation of the new line
We know that parallel lines have the same slope. Therefore, the line we are looking for has a slope of 4 and must pass through the point [tex]\((3, 9)\)[/tex].
We use the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the point the line passes through and [tex]\(m\)[/tex] is the slope. Plugging in the known values:
[tex]\[ y - 9 = 4(x - 3) \][/tex]
### Step 3: Simplify to obtain the slope-intercept form of the equation
Now, we simplify the equation to get it into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y - 9 = 4(x - 3) \][/tex]
[tex]\[ y - 9 = 4x - 12 \][/tex]
[tex]\[ y = 4x - 12 + 9 \][/tex]
[tex]\[ y = 4x - 3 \][/tex]
### Final Equation
The equation of the line that is parallel to [tex]\(4x - y = -6\)[/tex] and passes through the point [tex]\((3, 9)\)[/tex] is:
[tex]\[ y = 4x - 3 \][/tex]
### Step 1: Determine the slope of the given line
First, we will rewrite the equation of the given line [tex]\(4x - y = -6\)[/tex] in slope-intercept form [tex]\(y = mx + b\)[/tex]. This form makes it easier to identify the slope [tex]\(m\)[/tex].
Starting with the equation:
[tex]\[ 4x - y = -6 \][/tex]
Rearrange it to solve for [tex]\(y\)[/tex]:
[tex]\[ -y = -4x - 6 \][/tex]
[tex]\[ y = 4x + 6 \][/tex]
In the slope-intercept form [tex]\(y = mx + b\)[/tex], the coefficient of [tex]\(x\)[/tex] is the slope [tex]\(m\)[/tex]. Here, the slope [tex]\(m\)[/tex] is 4.
### Step 2: Use point-slope form to find the equation of the new line
We know that parallel lines have the same slope. Therefore, the line we are looking for has a slope of 4 and must pass through the point [tex]\((3, 9)\)[/tex].
We use the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the point the line passes through and [tex]\(m\)[/tex] is the slope. Plugging in the known values:
[tex]\[ y - 9 = 4(x - 3) \][/tex]
### Step 3: Simplify to obtain the slope-intercept form of the equation
Now, we simplify the equation to get it into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y - 9 = 4(x - 3) \][/tex]
[tex]\[ y - 9 = 4x - 12 \][/tex]
[tex]\[ y = 4x - 12 + 9 \][/tex]
[tex]\[ y = 4x - 3 \][/tex]
### Final Equation
The equation of the line that is parallel to [tex]\(4x - y = -6\)[/tex] and passes through the point [tex]\((3, 9)\)[/tex] is:
[tex]\[ y = 4x - 3 \][/tex]
