Question

Write an equation of line that passes through the points (4,3) land (1, 2) in point-slope form.

Answer 1

Answer:

$y=\frac{1}{3} x+\frac{5}{3}$

Step-by-step explanation:

Use the slope formula to find the slope:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{rise}{run}$

Insert the values:

$(4_{x_{1}},3_{y_{1}})\\\\(1_{x_{2}},2_{y_{2}})\\\\\frac{2-3}{1-4}=\frac{-1}{-3}=\frac{1}{3}$

The slope is 1/3.

You can do this two ways. You can solve using slope-intercept and solve for b, or  use point-slope form, then simplify to slope-intercept

Use the slope-intercept form:

$y=mx+b$

where:

• m is the slope
• b is the y-intercept
• x and y are corresponding coordinate points (x,y)

Insert the slope and one of the given coordinate points:

$(1_{x},2_{y})\\\\2=\frac{1}{3} (1)+b$

Solve for the y-intercept. Simplify:

$2=\frac{1}{3} +b$

Subtract 1/3 from both sides:

$2-\frac{1}{3} =\frac{1}{3} -\frac{1}{3} +b\\\\2-\frac{1}{3} =b$

Subtract. Find the common denominator:

$\frac{6}{3} -\frac{1}{3} =\frac{5}{3}$

Insert:

$\frac{5}{3} =b$

The y-intercept is 5/3. Insert:

$y=\frac{1}{3}x+\frac{5}{3}$

or

Use point-slope form:

$y-y_{1}=m(x-x_{1})$

where:

• m is the slope
• x1 and y1 are corresponding coordinate points

Insert values:

$(1_{x_{1}},2_{y_{1}})\\\\y-2=\frac{1}{3} (x-1)$

Simplify the equation by solving for y. This will result in the slope-intercept form. Simplify multiplication using the rule $\frac{a}{b} *c=\frac{ac}{b}$:

$y-2=\frac{1(x-1)}{3} \\\\y-2=\frac{x-1}{3}$

Isolate the variable. Add 2 to both sides:

$y-2+2=\frac{x-1}{3} +2\\\\y=\frac{x-1}{3}+2$

Simplify to slope-intercept form, y=mx+b. Use the rule $\frac{a-b}{c} =\frac{a}{c}-\frac{b}{c}$:

$y=\frac{x}{3} -\frac{1}{3} +2$

Combine like terms. Find a common denominator:

$-\frac{1}{3}+2=-\frac{1}{3}+\frac{2}{1}\\\\-\frac{1}{3}+\frac{3*2}{3*1}=-\frac{1}{3}+\frac{6}{3} \\\\-\frac{1}{3}+\frac{6}{3} =\frac{5}{3}$

Insert:

$y=\frac{x}{3} +\frac{5}{3}$

Use the rule $\frac{x}{a} =\frac{1}{a}x$:

$y=\frac{1}{3} x+\frac{5}{3}$

:Done