The linear equality represented by the graphis [tex]\boxed{{\mathbf{y > 3x + 2}}}[/tex] and it matches with [tex]\boxed{{\mathbf{OPTION B}}}[/tex].
Further explanation:
It is given that a line passes through points [tex]\left( {0,2} \right)[/tex] and [tex]\left({ - 3, - 7}\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=\dfrac{{{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]2[/tex] for [tex]{y_1}[/tex] , [tex]-3[/tex] for [tex]{x_2}[/tex] and [tex]-7[/tex] for [tex]{y_2}[/tex] in equation (1) to obtain the slope of a line that passes through points [tex]\left({0,2}\right)[/tex] and [tex]\left({ - 3, - 7}\right)[/tex].
[tex]\begin{aligned}m&=\frac{{ - 7 - 2}}{{ - 3 - 0}}\\&=\frac{{ - 9}}{{ - 3}}\\&=3\\\end{aligned}[/tex]
Therefore, the slope is [tex]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]2[/tex] for [tex]{y_1}[/tex] and [tex]3[/tex] for [tex]m[/tex] in equation (2) to obtain the equation of line.
[tex]\begin{aligned}y - 2&=3\left({x - 0}\right)\\y - 2&=3x\\y&=3x + 2\\\end{aligned}[/tex]
Therefore, the value of [tex]y[/tex] is [tex]3x + 2[/tex].
Since the shaded part in Figure 1 is above the equation of line [tex]y = 3x + 2[/tex], therefore, greater than sign is used instead of is equal to.
Thus, the linear inequality is [tex]y > 3x + 2[/tex] as shown below in Figure 2.
Now, the four options are given below.
[tex]\begin{aligned}{\text{OPTION A}} \to y < 3x + 2 \hfill\\{\text{OPTION B}} \to y > 3x + 2 \hfill\\{\text{OPTION C}} \to y < x + 2 \hfill\\{\text{OPTION D}} \to y > x + 2 \hfill\\\end{aligned}[/tex]
Since OPTION B matches the obtained equation that is [tex]y > 3x + 2[/tex].
Thus, the linear equality represented by the graph is [tex]\boxed{{\mathbf{y > 3x + 2}}}[/tex] and it matches with [tex]\boxed{{\mathbf{OPTION B}}}[/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,equation of line, line, passes through point, inequality
