Question

Graph the system of equations below. What is the solution?

y=2x-3
y=-x+3

Answer 1

To graph the system of equations given by:

1. \( y = 2x - 3 \)
2. \( y = -x + 3 \)

we can find the intersection point of these two lines, which will give us the solution to the system.

### Step 1: Graphing the equations

**Equation 1: \( y = 2x - 3 \)**

- This is a straight line with a slope of 2 and a y-intercept of -3.
- To graph it, we can find a couple of points:
- When \( x = 0 \): \( y = 2(0) - 3 = -3 \) (point: \( (0, -3) \))
- When \( x = 1 \): \( y = 2(1) - 3 = -1 \) (point: \( (1, -1) \))

Thus, we can plot the points \( (0, -3) \) and \( (1, -1) \).

**Equation 2: \( y = -x + 3 \)**

- This is also a straight line with a slope of -1 and a y-intercept of 3.
- To graph it, we find a couple of points:
- When \( x = 0 \): \( y = -0 + 3 = 3 \) (point: \( (0, 3) \))
- When \( x = 3 \): \( y = -3 + 3 = 0 \) (point: \( (3, 0) \))

We can plot the points \( (0, 3) \) and \( (3, 0) \).

### Step 2: Finding the intersection

To find the solution, we set the two equations equal to each other to find the x-coordinate of the intersection point:

\[
2x - 3 = -x + 3
\]

### Step 3: Solve for \( x \)

1. Add \( x \) to both sides:
\[
2x + x - 3 = 3
\]
\[
3x - 3 = 3
\]

2. Add 3 to both sides:
\[
3x = 6
\]

3. Divide by 3:
\[
x = 2
\]

### Step 4: Find \( y \)

Now substitute \( x = 2 \) back into either equation to find \( y \). Using the first equation:

\[
y = 2(2) - 3 = 4 - 3 = 1
\]

### Solution

Thus, the solution to the system of equations is:

\[
(x, y) = (2, 1)
\]

### Conclusion

The intersection point of the lines \( y = 2x - 3 \) and \( y = -x + 3 \) is \( (2, 1) \). This point is the solution to the system of equations.