The linear equality represented by the graphis \(\boxed{{\mathbf{y > 3x + 2}}}\) and it matches with \(\boxed{{\mathbf{OPTION B}}}\).
Further explanation:
It is given that a line passes through points \(\left( {0,2} \right)\) and \(\left({ - 3, - 7}\right)\) as shown below in Figure 1
The slope of a line passes through points \(\left({{x_1},{y_1}}\right)\) and \(\left({{x_2},{y_2}}\right)\) is calculated as follows:
\(m=\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\) ......(1)
Here, the slope of a line is denoted as \(m\) and points are \(\left({{x_1},{y_1}}\right)\) and \(\left({{x_2},{y_2}}\right)\).
Substitute \(0\) for \({x_1}\) , \(2\) for \({y_1}\) , \(-3\) for \({x_2}\) and \(-7\) for \({y_2}\) in equation (1) to obtain the slope of a line that passes through points \(\left({0,2}\right)\) and \(\left({ - 3, - 7}\right)\).
\(\begin{aligned}m&=\frac{{ - 7 - 2}}{{ - 3 - 0}}\\&=\frac{{ - 9}}{{ - 3}}\\&=3\\\end{aligned}\)
Therefore, the slope is \(3\).
The point-slope form of the equation of a line with slope \(m\) passes through point \(\left({{x_1},{y_1}}\right)\) is represented as follows:
\(y - {y_1}= m\left({x - {x_1}}\right)\) ......(2)
Substitute \(0\) for \({x_1}\) , \(2\) for \({y_1}\) and \(3\) for \(m\) in equation (2) to obtain the equation of line.
\(\begin{aligned}y - 2&=3\left({x - 0}\right)\\y - 2&=3x\\y&=3x + 2\\\end{aligned}\)
Therefore, the value of \(y\) is \(3x + 2\).
Since the shaded part in Figure 1 is above the equation of line \(y = 3x + 2\), therefore, greater than sign is used instead of is equal to.
Thus, the linear inequality is \(y > 3x + 2\) as shown below in Figure 2.
Now, the four options are given below.
\(\begin{aligned}{\text{OPTION A}} \to y < 3x + 2 \hfill\\{\text{OPTION B}} \to y > 3x + 2 \hfill\\{\text{OPTION C}} \to y < x + 2 \hfill\\{\text{OPTION D}} \to y > x + 2 \hfill\\\end{aligned}\)
Since OPTION B matches the obtained equation that is \(y > 3x + 2\).
Thus, the linear equality represented by the graph is \(\boxed{{\mathbf{y > 3x + 2}}}\) and it matches with \(\boxed{{\mathbf{OPTION B}}}\).
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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
