Answer:
[tex]y=\frac{1}{3} x+\frac{5}{3}[/tex]
Step-by-step explanation:
Use the slope formula to find the slope:
[tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{rise}{run}[/tex]
Insert the values:
[tex](4_{x_{1}},3_{y_{1}})\\\\(1_{x_{2}},2_{y_{2}})\\\\\frac{2-3}{1-4}=\frac{-1}{-3}=\frac{1}{3}[/tex]
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:
[tex]y=mx+b[/tex]
where:
Insert the slope and one of the given coordinate points:
[tex](1_{x},2_{y})\\\\2=\frac{1}{3} (1)+b[/tex]
Solve for the y-intercept. Simplify:
[tex]2=\frac{1}{3} +b[/tex]
Subtract 1/3 from both sides:
[tex]2-\frac{1}{3} =\frac{1}{3} -\frac{1}{3} +b\\\\2-\frac{1}{3} =b[/tex]
Subtract. Find the common denominator:
[tex]\frac{6}{3} -\frac{1}{3} =\frac{5}{3}[/tex]
Insert:
[tex]\frac{5}{3} =b[/tex]
The y-intercept is 5/3. Insert:
[tex]y=\frac{1}{3}x+\frac{5}{3}[/tex]
or
Use point-slope form:
[tex]y-y_{1}=m(x-x_{1})[/tex]
where:
Insert values:
[tex](1_{x_{1}},2_{y_{1}})\\\\y-2=\frac{1}{3} (x-1)[/tex]
Simplify the equation by solving for y. This will result in the slope-intercept form. Simplify multiplication using the rule [tex]\frac{a}{b} *c=\frac{ac}{b}[/tex]:
[tex]y-2=\frac{1(x-1)}{3} \\\\y-2=\frac{x-1}{3}[/tex]
Isolate the variable. Add 2 to both sides:
[tex]y-2+2=\frac{x-1}{3} +2\\\\y=\frac{x-1}{3}+2[/tex]
Simplify to slope-intercept form, y=mx+b. Use the rule [tex]\frac{a-b}{c} =\frac{a}{c}-\frac{b}{c}[/tex]:
[tex]y=\frac{x}{3} -\frac{1}{3} +2[/tex]
Combine like terms. Find a common denominator:
[tex]-\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}[/tex]
Insert:
[tex]y=\frac{x}{3} +\frac{5}{3}[/tex]
Use the rule [tex]\frac{x}{a} =\frac{1}{a}x[/tex]:
[tex]y=\frac{1}{3} x+\frac{5}{3}[/tex]
:Done
