To find the equation of a line that passes through the point [tex]\((4,2)\)[/tex] and has a slope of [tex]\(\frac{5}{4}\)[/tex], we'll use the slope-intercept form of a line, which is:
[tex]\[ y = mx + b \][/tex]
Here, [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. We are given the slope [tex]\(m = \frac{5}{4}\)[/tex] and the point [tex]\((x_1, y_1) = (4, 2)\)[/tex].
Step 1: Substitute the slope [tex]\(m\)[/tex] and the coordinates of the point into the slope-intercept form equation to solve for [tex]\(b\)[/tex].
[tex]\[ y_1 = mx_1 + b \][/tex]
Step 2: Plugging in the values:
[tex]\[ 2 = \left(\frac{5}{4}\right) (4) + b \][/tex]
Step 3: Simplify the equation to isolate [tex]\(b\)[/tex]:
[tex]\[ 2 = 5 + b \][/tex]
[tex]\[ b = 2 - 5 \][/tex]
[tex]\[ b = -3 \][/tex]
Step 4: Now that we have the slope [tex]\(m = \frac{5}{4}\)[/tex] and the y-intercept [tex]\(b = -3\)[/tex], we can write the equation of the line:
[tex]\[ y = \frac{5}{4}x - 3 \][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[ y = 1.25x - 3 \][/tex]
