Question

Graph the system below and write its solution. 1 y=x+1 x-2y=-2 Note that you can also answer "No solution" or "Infinitely many" solutions.

Answer 1

To find the solution of the system of equations given:

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

First, let's graph these equations to visualize the solution.

### Step 1: Graph the first equation \( y = x + 1 \)

This is a linear equation in slope-intercept form (y = mx + b) where:
- The slope (m) is 1
- The y-intercept (b) is 1

Using this information:
- When \( x = 0 \), \( y = 1 \) (point (0, 1))
- When \( x = -1 \), \( y = 0 \) (point (-1, 0))
- When \( x = 1 \), \( y = 2 \) (point (1, 2))

### Step 2: Graph the second equation \( x - 2y = -2 \)

We can convert this to slope-intercept form (y = mx + b) as follows:

1. Rearrange the equation:
\[
x + 2 = 2y
\]
\[
y = \frac{1}{2}x + 1
\]

Now we can see the slope is \( \frac{1}{2} \) and the y-intercept is 1.

Using this information:
- When \( x = 0 \), \( y = 1 \) (point (0, 1))
- When \( x = 2 \), \( y = 2 \) (point (2, 2))
- When \( x = -2 \), \( y = 0 \) (point (-2, 0))

### Step 3: Plot both lines on a graph

- The first line \( y = x + 1 \) passes through (0, 1) and has a slope of 1.
- The second line \( y = \frac{1}{2}x + 1 \) also passes through (0, 1) but with a gentler slope.

### Step 4: Find the intersection

Notice that both equations have a y-intercept of 1. This means that they intersect at the point (0, 1). We should confirm if they intersect elsewhere:

To find the intersection algebraically:
1. Substitute \( y = x + 1 \) into the second equation:
\[
x - 2(x + 1) = -2
\]
Simplifying:
\[
x - 2x - 2 = -2 \quad \Rightarrow \quad -x - 2 = -2 \quad \Rightarrow \quad -x = 0 \quad \Rightarrow \quad x = 0
\]
Substituting back into the first equation:
\[
y = 0 + 1 = 1
\]
Thus, the solution is the point (0, 1).

### Conclusion

The intersection point (which is the solution to the system of equations) is:

\[
\boxed{(0, 1)}
\]

### Summary

The lines intersect at the point (0, 1), hence, the system has a **unique solution** at this point.