Respuesta :

InesWalston

Answer-

[tex]\boxed{\boxed{y > 3x + 2}}[/tex]

Solution-

From the graph, the coordinates of two present on the line are, (0, 2), (-3, -7)

Applying two point formula for straight line,

[tex]\Rightarrow \dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}[/tex]

Here,

x₁ = 0

y₁ = 2

x₂ = -3

y₂ = -7

Putting in the formula,

[tex]\Rightarrow \dfrac{y-2}{-7-2}=\dfrac{x-0}{-3-0}[/tex]

[tex]\Rightarrow \dfrac{y-2}{-9}=\dfrac{x}{-3}[/tex]

[tex]\Rightarrow \dfrac{y-2}{3}=\dfrac{x}{1}[/tex]

[tex]\Rightarrow \dfrac{y-2}{3}=x[/tex]

[tex]\Rightarrow {y-2}=3x[/tex]

[tex]\Rightarrow y=3x+2[/tex]

As we have got the equality equation, now we have to care about the sign.

So option 3 and 4 are not the correct options.

Putting origin (0, 0) in the first equation i.e [tex]y < 3x + 2[/tex],

[tex]\Rightarrow 0<3(0)+2\\\\\Rightarrow 0<2[/tex]

As this satisfies the equation the shaded region will be towards origin or right side of the line [tex]y=3x+2[/tex]. This area is the opposite to the desired area, so this is not the right answer.

Putting origin (0, 0) in the second equation i.e [tex]y > 3x + 2[/tex],

[tex]\Rightarrow 0>3(0)+2\\\\\Rightarrow 0\ngtr 2[/tex]

As this does not satisfy the equation the shaded region will be opposite to origin or left side of the line [tex]y=3x+2[/tex], so this is the right answer.

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

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