Question

Consider the two equations:

2y-2x=10

2y+2x=10

Select four statements that are true for the equations.



a
The system of equations has a solution.

b
The graph of equations intersects at a point.

c
The graph solution is at (5,0)

d
The graph intersects at (0,5)

e

The intersection point is the solution of the system of equations.

f
There are two solutions

Answer 1

To analyze the two equations given:

1) \( 2y - 2x = 10 \)
2) \( 2y + 2x = 10 \)

Let's rewrite these equations in slope-intercept form (\(y = mx + b\)).

For the first equation:
\[
2y - 2x = 10 \implies 2y = 2x + 10 \implies y = x + 5
\]

For the second equation:
\[
2y + 2x = 10 \implies 2y = -2x + 10 \implies y = -x + 5
\]

Now, we can analyze the geometrical implications of these equations:

1. The first equation \(y = x + 5\) describes a line with slope 1, which crosses the y-axis at (0,5).
2. The second equation \(y = -x + 5\) describes a line with slope -1, which also crosses the y-axis at (0,5).

Since both lines cross the y-axis at the same point (0,5) and have different slopes, they will intersect only at this point.

Now let’s evaluate the statements:

a. **The system of equations has a solution.**
- **True.** The two lines intersect at the point (0, 5), so there is one solution.

b. **The graph of equations intersects at a point.**
- **True.** The lines intersect at (0, 5).

c. **The graph solution is at (5,0).**
- **False.** The solution is at (0, 5), not (5, 0).

d. **The graph intersects at (0, 5).**
- **True.** This is where they intersect.

e. **The intersection point is the solution of the system of equations.**
- **True.** The intersection point (0, 5) is indeed the solution.

f. **There are two solutions.**
- **False.** There is exactly one solution, (0, 5).

Thus, the four true statements are:

- a
- b
- d
- e