To determine the equation of a line that passes through the point [tex]\((4,3)\)[/tex] and has a slope of 2, we can use the point-slope form of a linear equation. The point-slope form 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. Here, [tex]\((x_1, y_1) = (4, 3)\)[/tex] and [tex]\(m = 2\)[/tex].
Let's substitute the given values into the point-slope form equation:
[tex]\[ y - 3 = 2(x - 4) \][/tex]
Next, we will simplify this equation to get it into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
1. Expand the right-hand side:
[tex]\[ y - 3 = 2x - 8 \][/tex]
2. Add 3 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = 2x - 8 + 3 \][/tex]
3. Simplify the right-hand side:
[tex]\[ y = 2x - 5 \][/tex]
Thus, the equation in slope-intercept form is [tex]\(y = 2x - 5\)[/tex].
Let's now check which of the given choices matches this equation:
- [tex]\(y = 2x - 1\)[/tex]
- [tex]\(y = 2x - 5\)[/tex]
- [tex]\(y = 2x - 11\)[/tex]
- [tex]\(y = 2x - 7\)[/tex]
The correct equation is:
[tex]\[ y = 2x - 5 \][/tex]
Therefore, the equation of the line that passes through the point [tex]\((4,3)\)[/tex] and has a slope of 2 is:
[tex]\[ y = 2x - 5 \][/tex]
