what is the intersection of y=3-x and y=2x

We are given two equations:

\[
y = 3 - x \tag{1}
\]
\[
y = 2x \tag{2}
\]

To find the point of intersection, we need to solve this system of equations by setting the right-hand sides equal to each other (since both equal \(y\)):

\[
3 - x = 2x
\]

### Step 1: Solve for \(x\)
Add \(x\) to both sides:

\[
3 = 3x
\]

Now, divide by 3:

\[
x = 1
\]

### Step 2: Solve for \(y\)
Substitute \(x = 1\) into either equation. Let's use equation \((2)\):

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

### Final solution:
The point of intersection is:
\[
(x, y) = (1, 2)
\]

Thus, the lines intersect at the point \((1, 2)\).

Would you like to explore this concept further?

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Here are 8 related questions for further exploration:
1. What happens if you change the equations to \(y = 3x\) and \(y = 5 - x\)?
2. How would you find the intersection point using substitution or elimination methods?
3. Can these equations be solved graphically? What does the graph look like?
4. How would the intersection change if one equation was \(y = 2x + 1\)?
5. What is the geometric interpretation of this solution?
6. What if you had a system of three linear equations—how would you find their intersection?
7. How would you find the intersection of a linear equation and a quadratic equation?
8. Can you solve this system using matrix methods?

**Tip**: Plotting the equations on a graph can give a visual representation of where the lines intersect.

Learn how to find the intersection point of y=3-x and y=2x with a step-by-step solution. Understand the concept of intersecting linear equations and solve them using algebraic methods. Suitable for students in Grades 7-9.

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