Question

Graph the system of equations on a grid and point of intersection point c
y=3x+3
y=x+5

Answer 1

To graph the system of equations \( y = 3x + 3 \) and \( y = x + 5 \) and find the point of intersection, we can follow these steps:

### Step 1: Graph the equations

**Equation 1: \( y = 3x + 3 \)**
- This is a linear equation with a slope of 3 and a y-intercept (where x = 0) at 3.
- To plot it, we can find a couple of points:
- When \( x = 0 \):
\[
y = 3(0) + 3 = 3 \quad \Rightarrow \quad (0, 3)
\]
- When \( x = 1 \):
\[
y = 3(1) + 3 = 6 \quad \Rightarrow \quad (1, 6)
\]
- When \( x = -1 \):
\[
y = 3(-1) + 3 = 0 \quad \Rightarrow \quad (-1, 0)
\]

**Equation 2: \( y = x + 5 \)**
- This linear equation has a slope of 1 and a y-intercept at 5.
- To plot it, we can find a couple of points:
- When \( x = 0 \):
\[
y = 0 + 5 = 5 \quad \Rightarrow \quad (0, 5)
\]
- When \( x = 1 \):
\[
y = 1 + 5 = 6 \quad \Rightarrow \quad (1, 6)
\]
- When \( x = -1 \):
\[
y = -1 + 5 = 4 \quad \Rightarrow \quad (-1, 4)
\]

### Step 2: Plot the points

Now, we can plot the points we found for both equations:

- For \( y = 3x + 3 \): Points (0, 3), (1, 6), and (-1, 0).
- For \( y = x + 5 \): Points (0, 5), (1, 6), and (-1, 4).

You can draw the lines based on these points.

### Step 3: Find the point of intersection

To find the point where the two lines intersect, set the equations equal to each other:
\[
3x + 3 = x + 5
\]
Now, solve for \( x \):
\[
3x - x = 5 - 3 \\
2x = 2 \\
x = 1
\]
Now substitute \( x = 1 \) back into either equation to find \( y \):
\[
y = 3(1) + 3 = 6 \quad \text{or} \quad y = 1 + 5 = 6
\]

### Conclusion

The point of intersection (labeled point \( C \)) is:
\[
C = (1, 6)
\]

So, we have graphed the two lines, and they intersect at point \( C(1, 6) \).