Answer :
Sure, let's find the equation of the line that passes through the point [tex]\((-1, 2)\)[/tex] and is parallel to the line [tex]\(y = x + 4\)[/tex].
### Step-by-Step Solution:
1. Determine the Slope of the Given Line
The given line is [tex]\(y = x + 4\)[/tex]. This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope and [tex]\(b\)[/tex] is the y-intercept.
Here, [tex]\(m = 1\)[/tex] because the coefficient of [tex]\(x\)[/tex] is 1.
2. Recognize that Parallel Lines Have the Same Slope
Since the new line we are finding is parallel to [tex]\(y = x + 4\)[/tex], it will have the same slope. Therefore, the slope [tex]\(m\)[/tex] of our new line is also 1.
3. Use the Point-Slope Form of a Line Equation
The point-slope form of a line equation is given by:
[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.
Substituting the given point [tex]\((-1, 2)\)[/tex] and the slope [tex]\(m = 1\)[/tex] into the point-slope form:
[tex]\[ y - 2 = 1(x - (-1)) \][/tex]
[tex]\[ y - 2 = 1(x + 1) \][/tex]
4. Simplify the Equation
Distribute the 1 on the right-hand side:
[tex]\[ y - 2 = x + 1 \][/tex]
Add 2 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = x + 3 \][/tex]
Thus, the equation of the line that passes through the point [tex]\((-1, 2)\)[/tex] and is parallel to the line [tex]\(y = x + 4\)[/tex] is:
[tex]\[ \boxed{y = x + 3} \][/tex]
### Step-by-Step Solution:
1. Determine the Slope of the Given Line
The given line is [tex]\(y = x + 4\)[/tex]. This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope and [tex]\(b\)[/tex] is the y-intercept.
Here, [tex]\(m = 1\)[/tex] because the coefficient of [tex]\(x\)[/tex] is 1.
2. Recognize that Parallel Lines Have the Same Slope
Since the new line we are finding is parallel to [tex]\(y = x + 4\)[/tex], it will have the same slope. Therefore, the slope [tex]\(m\)[/tex] of our new line is also 1.
3. Use the Point-Slope Form of a Line Equation
The point-slope form of a line equation is given by:
[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.
Substituting the given point [tex]\((-1, 2)\)[/tex] and the slope [tex]\(m = 1\)[/tex] into the point-slope form:
[tex]\[ y - 2 = 1(x - (-1)) \][/tex]
[tex]\[ y - 2 = 1(x + 1) \][/tex]
4. Simplify the Equation
Distribute the 1 on the right-hand side:
[tex]\[ y - 2 = x + 1 \][/tex]
Add 2 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = x + 3 \][/tex]
Thus, the equation of the line that passes through the point [tex]\((-1, 2)\)[/tex] and is parallel to the line [tex]\(y = x + 4\)[/tex] is:
[tex]\[ \boxed{y = x + 3} \][/tex]
