Question

Question 3: No Solution%0D%0AConsider the two equations you put a square around, which form a system with no %0D%0Asolution. Graph them together on the following coordinate grid. Plug a few points into the original equations and check to show that the system of %0D%0Aequations has no solution. Explain your work.

Answer 1

[AI-generated answer — Bot GPT 3.5]

Since there is no solution to the system of equations, our two equations are likely parallel lines. Let's take the following equations from the previous example:

1) 2x + 3y = 7
2) 2x + 3y = 10

Let's rearrange these equations to slope-intercept form (y = mx + b):

1) 3y = -2x + 7
y = (-2/3)x + 7/3

2) 3y = -2x + 10
y = (-2/3)x + 10/3

Now let's plot these two lines on a coordinate grid:

Equation 1: y = (-2/3)x + 7/3
- At x = 0, y = 7/3 ≈ 2.33
- At x = 3, y = 0
- Line passes through (0, 7/3) and (3, 0)

Equation 2: y = (-2/3)x + 10/3
- At x = 0, y = 10/3 ≈ 3.33
- At x = 3, y = 2
- Line passes through (0, 10/3) and (3, 2)

Both lines have the same slope (-2/3) but different y-intercepts. Therefore, they are parallel lines that will never intersect, indicating no solution to the system of equations.

To confirm this mathematically, let's substitute a point from one equation into the other:

Using equation 1 (2x + 3y = 7):
- Let x = 0
- 3y = 7
- y = 7/3 ≈ 2.33

Substitute this y value into equation 2 (2x + 3y = 10):
- 2(0) + 3(7/3) = 10
- 7 = 10 (which is false)

Therefore, the system of equations has no solution, as the lines are parallel and do not intersect.