Question

Samantha is solving the following system of equations using the method of substitution: 3x + 2y = 11 - x + y = 3 The following are Samantha's first two steps. y= x + 3 3x + 2(x+3) =11 HINT: Solve the bottom equation for x and then plug that value back into the first equation to solve for y. Arrange the steps below to match the order you used to do this.

Place each of the remaining steps in the appropriate order to finish solving the system of equations.

1 5x =5
2 (1,4)
3 y=4
4 x=1
5 3x + 2x + 6 = 11
6 5x + 6 = 11
7 y= 1 + 3

Which ordered pair is the solution to the system of linear equations displayed on the graph below?

HINT: The solution is the point where the two lines intersect. What is the x coordinate and y coordinate at that point of intersection?

A:
`(0,-(3)/(2) )`
B:
`(-(3)/(2),0)`
C:
`(0,(3)/(2)`

Which of the following inequalities can be represented by the graph below?

HINT: Can you find the equation for the boundary line? What is its y-intercept? What is its slope? Note that the region above the line is shaded.

A:
`y\geq2x-2`
B:
`y\geq (1)/(2)x-2`
C:
`y\geq (1)/(2)x +2`

Answer 1

According to the numbers provided, the sequence is:

5, 6, 1, 4, 7, 3, 2

Solution of the two lines intersecting in the graph is answer B

The inequality that represents the graph is that of answer A

Explanation:

First problem:

This is the sequence to solve for "x" once the substitution was made;

`3x +2\,(x+3)=11\\3x+2x+6=11\\5x+6=11\\5x=11-6\\5x=5\\x=1\\`

Now that we know x, we use its value (1) in the substitution equation in order to find "y" and complete the ordered pair:

`y=x+3\\y=1+3\\y=4`

Then the solution pair (x, y) is: (1, 4)

Therefore, according to the numbers provided in the list of possible steps, the order is:

5, 6, 1, 4, 7, 3, 2

Second problem:

The solution to the system is where the two lines two lines intersect: "the point -1.5 on the x axis. Notice that -1.5 can be written in fraction form as -3/2, this is the x-coordinate of the intersection point. The y-coordinate is y-0 since the point is on the x-axis. Therefore the point is (-3/2, 0) which is given by answer B).

Third problem: The boundary line seems to go through the points: (-4, -10) and (0, -2), so we can calculate the slope via the equation:

`slope=(y_2-y_1)/(x_2-x_1) \\slope=(-2-(-10))/(0-(-4))\\slope=(8)/(4)\\slope=2`

since the line's y-intercept is (0, -2), then the equation of the boundary line in slope-intercept is:

`y=2x-2`

Since the shaded region includes the boundary line and the zone above it (with y-values larger than or equal to those point in the line), we can say that the inequality that represents this graph is:

`y\geq 2x-2`

which corresponds to answer A