Answer :
To determine the equation of a line in slope-intercept form that passes through the point (3, -2) and is parallel to the line [tex]\(y = 2x + 4\)[/tex], we need to follow these steps:
### Step 1: Identify the slope of the given line
The given line is [tex]\(y = 2x + 4\)[/tex]. In this form, [tex]\(y = mx + b\)[/tex], [tex]\(m\)[/tex] represents the slope, and [tex]\(b\)[/tex] is the y-intercept. Here, the slope ([tex]\(m\)[/tex]) is 2.
### Step 2: Understand that parallel lines have the same slope
Since we need a line that is parallel to [tex]\(y = 2x + 4\)[/tex], it means our new line will have the same slope, which is 2.
### Step 3: Use the point-slope form of the line equation
The point-slope form of the line equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here:
- [tex]\(m\)[/tex] = 2 (the slope we identified)
- [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes, which is (3, -2).
Substitute these values into the point-slope form:
[tex]\[ y - (-2) = 2(x - 3) \][/tex]
### Step 4: Simplify the equation
Simplify the left side of the equation:
[tex]\[ y + 2 = 2(x - 3) \][/tex]
Next, distribute the slope (2) on the right side:
[tex]\[ y + 2 = 2x - 6 \][/tex]
Then, isolate [tex]\(y\)[/tex] by subtracting 2 from both sides:
[tex]\[ y = 2x - 6 - 2 \][/tex]
[tex]\[ y = 2x - 8 \][/tex]
### Step 5: Write the final equation in slope-intercept form
The equation of the line that passes through (3, -2) and is parallel to [tex]\(y = 2x + 4\)[/tex] is:
[tex]\[ \boxed{y = 2x - 8} \][/tex]
This is the equation in slope-intercept form that meets the given criteria.
### Step 1: Identify the slope of the given line
The given line is [tex]\(y = 2x + 4\)[/tex]. In this form, [tex]\(y = mx + b\)[/tex], [tex]\(m\)[/tex] represents the slope, and [tex]\(b\)[/tex] is the y-intercept. Here, the slope ([tex]\(m\)[/tex]) is 2.
### Step 2: Understand that parallel lines have the same slope
Since we need a line that is parallel to [tex]\(y = 2x + 4\)[/tex], it means our new line will have the same slope, which is 2.
### Step 3: Use the point-slope form of the line equation
The point-slope form of the line equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here:
- [tex]\(m\)[/tex] = 2 (the slope we identified)
- [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes, which is (3, -2).
Substitute these values into the point-slope form:
[tex]\[ y - (-2) = 2(x - 3) \][/tex]
### Step 4: Simplify the equation
Simplify the left side of the equation:
[tex]\[ y + 2 = 2(x - 3) \][/tex]
Next, distribute the slope (2) on the right side:
[tex]\[ y + 2 = 2x - 6 \][/tex]
Then, isolate [tex]\(y\)[/tex] by subtracting 2 from both sides:
[tex]\[ y = 2x - 6 - 2 \][/tex]
[tex]\[ y = 2x - 8 \][/tex]
### Step 5: Write the final equation in slope-intercept form
The equation of the line that passes through (3, -2) and is parallel to [tex]\(y = 2x + 4\)[/tex] is:
[tex]\[ \boxed{y = 2x - 8} \][/tex]
This is the equation in slope-intercept form that meets the given criteria.
