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Math  /  Algebra

Questiona. (3,4)(3,4) and (6,5)(6,5) where slope =13=\frac{1}{3}
The equation of the line is □\square (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Studdy Solution

STEP 1

1. We are given two points on a line: (3,4)(3,4) and (6,5)(6,5).
2. The slope of the line is 13\frac{1}{3}.
3. We need to find the equation of the line in slope-intercept form, which is y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

STEP 2

1. Verify the given slope using the two points.
2. Use the point-slope form to find the equation of the line.
3. Convert the equation to slope-intercept form.

STEP 3

Verify the given slope using the formula for the slope between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2):
m=y2−y1x2−x1 m = \frac{y_2 - y_1}{x_2 - x_1}
Substitute the given points (3,4)(3,4) and (6,5)(6,5):
m=5−46−3=13 m = \frac{5 - 4}{6 - 3} = \frac{1}{3}
The given slope is correct.

STEP 4

Use the point-slope form of the equation of a line, which is:
y−y1=m(x−x1) y - y_1 = m(x - x_1)
Choose one of the given points, say (3,4)(3,4), and use the slope m=13m = \frac{1}{3}:
y−4=13(x−3) y - 4 = \frac{1}{3}(x - 3)

STEP 5

Simplify the equation:
y−4=13x−1 y - 4 = \frac{1}{3}x - 1
Add 4 to both sides to solve for yy:
y=13x+3 y = \frac{1}{3}x + 3
This is the equation of the line in slope-intercept form.
The equation of the line is:
y=13x+3 \boxed{y = \frac{1}{3}x + 3}

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