To graph the line with a slope of 2 passing through the point [tex]\((-1, 2)\)[/tex], follow these detailed steps:
1. Identify Key Information:
- Slope ([tex]\(m\)[/tex]): 2
- Point ([tex]\(x_1, y_1\)[/tex]): [tex]\((-1, 2)\)[/tex]
2. Use the Point-Slope Form of the Equation:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substitute the slope [tex]\((m = 2)\)[/tex] and the point [tex]\((-1, 2)\)[/tex] into this formula:
[tex]\[
y - 2 = 2(x - (-1))
\][/tex]
Simplify the equation:
[tex]\[
y - 2 = 2(x + 1)
\][/tex]
3. Convert to Slope-Intercept Form:
The slope-intercept form of a line is given by [tex]\(y = mx + b\)[/tex], where [tex]\(b\)[/tex] is the y-intercept.
Distribute the slope 2 on the right side of the equation:
[tex]\[
y - 2 = 2x + 2
\][/tex]
To solve for [tex]\(y\)[/tex], add 2 to both sides:
[tex]\[
y = 2x + 4
\][/tex]
4. Equation of the Line:
The equation of the line in slope-intercept form is:
[tex]\[
y = 2x + 4
\][/tex]
5. Graph the Line:
- Plot the Point [tex]\((-1, 2)\)[/tex]: Locate and mark the point [tex]\((-1, 2)\)[/tex] on the coordinate plane.
- Use the Slope: The slope is 2, which can be interpreted as "rise over run." This means for every 1 unit you move to the right on the x-axis, you move 2 units up on the y-axis. From the point [tex]\((-1, 2)\)[/tex]:
- Move right 1 unit to [tex]\((0, 2)\)[/tex] and then up 2 units to [tex]\((0, 4)\)[/tex]. Mark this new point.
- Draw the Line: Connect these points with a straight line extending in both directions.
In summary, the linear equation [tex]\(y = 2x + 4\)[/tex] represents the line with a slope of 2 passing through the point [tex]\((-1, 2)\)[/tex]. This is the line you will graph on the coordinate plane.
