Which system of equations does this graph represent?

a. y=-2
x-2y=6

b. y=-2
x+2y=6

c. x=-2
2x-y=3

d. x=-2
2x+y=-3

The system of equations is y = -2, x - 2y = 6 ⇒ (a)

Step-by-step explanation:

Let us revise some facts about the linear equation

The form of the linear equation is y = m x + b, where m is the slope of the line and b is the y-intercept [the line intersect the y-axis at point (0, b)]

The equation of any horizontal line is y = b

The equation of any vertical line is x = a, where (a, 0) is the x-intercept

Now let us study the given graph

→ Blue line

∵ The blue line is a horizontal line

∴ Its equation is y = b

∵ b is the y-coordinate of the y-intercept

→ The line intersects the y-axis at point (0, -2) which means b = -2

∴ b = -2

∴ The equation of the blue line is y = -2

We have two answers only with the equation y = -2, a and b, so the answer must be one of them. Let us find the equation of the red line to decide which one is the answer.

→ Red line

∵ The equation of the line is y = m x + b

∵ The read line intersects the y-axis at point (0, -3)

∴ b = -3

∴ y = m x + (-3)

∴ y = mx - 3

→ To find m substitute x and y by the coordinates of any point on the line

∵ Point (6, 0) lies on the line

∴ x = 6 and y = 0

∵ 0 = m(6) - 3

∴ 0 = 6m - 3

→ Add 3 to sides

∴ 0 + 3 = 6m - 3 + 3

∴ 3 = 6m

→ Divide both sides by 6 to find m

∴

∴

→ Substitute the value of m in the equation above

∴ y = x - 3

→ Let us put x and y on one side and the number on the other

   side by subtracting y and add 3 to both sides

∵ y - y + 3 = x - 3 + 3 - y

∴ 3 =   x - y

→ Multiply both sides by 2

∴ 2(3) = 2( x) - 2(y)

∴ 6 = x - 2y

→ Switch the two sides

∴ x - 2y = 6

∴ The equation of the red line is x - 2y = 6

∵ The equations of the blue line and the red line are in answer a

∴ The system of equations is y = -2, x - 2y = 6

Note: you can use the point of intersection of the lines on the graph (2, -2) and substitute its coordinates in each answer to find which answer will satisfy that the two sides in each equation are equal