The linear equality represented by the graph is [tex]\boxed{y\leqslant\frac{1}{3}x-1.3}[/tex].
Further explanation:
It is given that a line passes through points [tex]\left({0,-\,1.3}\right)[/tex] and [tex]\left({3,-\,0.3}\right)[/tex] as shown below in Figure 1.
The slope of a line passes through points [tex]\left({{x_1},{y_1}}\right)[/tex] and [tex]\left({{x_2},{y_2}}\right)[/tex] is calculated as follows:
[tex]m=\frac{{{y_2}-{y_1}}}{{{x_2}-{x_1}}}[/tex] .......(1)
Here, the slope of a line is denoted as [tex]m[/tex] and points are [tex]\left({{x_1},{y_1}}\right)[/tex] and [tex]\left({{x_2},{y_2}}\right)[/tex] .
Substitute [tex]0[/tex] for [tex]{x_1}[/tex] , [tex]-1.3[/tex] for [tex]{y_1}[/tex] , [tex]3[/tex] for [tex]{x_2}[/tex] and [tex]-0.3[/tex] for [tex]{y_2}[/tex] in equation (1) to obtain the slope of a line that passes through points [tex]\left({0,-\,1.3}\right)[/tex] and [tex]\left({3,-\,0.3}\right)[/tex] .
[tex]\begin{aligned}m&=\frac{{-0.3-\left({-1.3}\right)}}{{3-0}}\\&=\frac{1}{3}\\\end{aligned}[/tex]
Therefore, the slope is [tex]\frac{1}{3}[/tex] .
The point-slope form of the equation of a line with slope [tex]m[/tex] passes through point [tex]\left({{x_1},{y_1}}\right)[/tex] is represented as follows:
[tex]y-{y_1}=m\left({x-{x_1}}\right)[/tex] ......(2)
Substitute [tex]0[/tex] for [tex]{x_1}[/tex] , [tex]-1.3[/tex] for [tex]{y_1}[/tex] and [tex]\frac{1}{3}[/tex] for [tex]m[/tex] in equation (2) to obtain the equation of line.
[tex]\begin{aligned}y-\left({-1.3}\right)&=\frac{1}{3}\left({x-0}\right)\\y+1.3&=\frac{1}{3}x\\y&=\frac{1}{3}x-1.3\\\end{aligned}[/tex]
Therefore, the value of [tex]y[/tex] is [tex]\frac{1}{3}x-1.3[/tex] .
Since the shaded part in Figure 1 is below the equation of line [tex]y=\frac{1}{3}x-1.3[/tex] , therefore, less than sign is also used with “is equal to”.
Thus, the linear inequality is [tex]y\leqslant\frac{1}{3}x-1.3[/tex] .
Thus, the linear equality represented by the graph is [tex]\boxed{y\leqslant\frac{1}{3}x-1.3}[/tex] .
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Answer Details:
Grade: Junior High School
Subject: Mathematics
Chapter: Coordinate Geometry
Keywords: Coordinate Geometry, linear equation, system of linear equations in two variables, variables, mathematics, inequality.
