Respuesta :
a. The point-slope equation is:
[tex]y-y_1=m(x-x_1)[/tex]Where m is the slope and (x1,y1) are the coordinates of one point in the line. Also, you need to write the equation with inequalities, then you need to replace the = sign, for a <, > or <=, >= sign.
Let's start by finding the slope of the first set of points (9,7) (8,5).
The formula for the slope is:
[tex]m=\frac{y_2-y_1}{x_2-x_1_{}}[/tex]By replacing the values you obtain:
[tex]m=\frac{5-7_{}}{8_{}-9}=\frac{-2}{-1}=2[/tex]The slope is 2.
Now, replace this value into the slope-form equation and the values of the first point (9,7):
[tex]y-7_{}>2(x-9)[/tex]I choose the sign > (greater than), but you can choose anyone, the difference will be for the solution of the inequality. When you solve the inequality you will find that the x-values have to be greater than the solution you found, or less than... etc, it will depend on the sign you have in the inequality.
b. The slope-intercept equation is:
[tex]y=mx+b[/tex]Where m is the slope and b the y-intercept.
Let's use the second set of points (2,9) and (2,7)
Start by calculating the slope:
[tex]m=\frac{7-9}{2-2}=\frac{-2}{0}=\text{ undefined}[/tex]As there's no difference in the x-coordinates, the line is a vertical line at x=2.
Also, there's no y-intercept as the line never crosses the y-axis.
I will use the first set again, so you can understand the slope-intercept form.
From part a) you know that the slope is 2, let's replace it in the equation and use the first pair of coordinates to find b:
[tex]\begin{gathered} 7=2\times9+b \\ 7=18+b \\ 7-18=b \\ b=-11 \end{gathered}[/tex]Thus, the slope-intercept with inequality will be:
[tex]y<2x-11[/tex]c. The standard form equation of a line is:
[tex]ax+by=c[/tex]Let's use the third set of points (3,5) (5,4).
Start by finding the slope:
[tex]m=\frac{4-5}{5-3}=\frac{-1}{2}=-0.5[/tex]Now, you can start with the point-slope form and then convert it into the standard form:
[tex]\begin{gathered} y-5\ge-0.5(x-3) \\ Apply\text{ the distributive property} \\ y-5\ge-0.5x+1.5 \\ y\ge-0.5x+1.5+5 \\ y\ge-0.5x+6.5 \\ 0.5x+y\ge6.5 \end{gathered}[/tex]Where a=0.5, b=1 and c=6.5
