At what point do the lines y=2x+3 and y=-x + 1 intersects

To find the point where the two lines \( y = 2x + 3 \) and \( y = -x + 1 \) intersect, we need to solve the system of equations. Since both equations are equal to \( y \), we can set them equal to each other:

\[
2x + 3 = -x + 1
\]

Now, solve for \( x \):

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

2. Subtract 3 from both sides:
\[
3x = 1 - 3
\]
\[
3x = -2
\]

3. Divide both sides by 3:
\[
x = -\frac{2}{3}
\]

Now that we have \( x = -\frac{2}{3} \), substitute it into one of the original equations to find \( y \). Let's use the first equation \( y = 2x + 3 \):

\[
y = 2\left(-\frac{2}{3}\right) + 3
\]
\[
y = -\frac{4}{3} + 3
\]
\[
y = -\frac{4}{3} + \frac{9}{3} = \frac{5}{3}
\]

Therefore, the lines intersect at the point \( \left( -\frac{2}{3}, \frac{5}{3} \right) \).

Would you like further clarification or more details?

Here are 5 related questions:
1. How do you determine if two lines are parallel or perpendicular?
2. What is the general method to find the intersection of two lines in 3D?
3. How can you find the distance between two non-intersecting lines?
4. Can you find the intersection point of lines given in parametric form?
5. What happens if two lines are coincident?

**Tip:** Always double-check your algebraic steps when solving systems of equations to avoid simple calculation errors.

At what point do the lines y=2x+3 and y=-x+1 intersect?

Learn how to find the intersection point of the lines y=2x+3 and y=-x+1 using algebraic methods. This step-by-step solution explains solving systems of linear equations, suitable for students in Grades 8-10.

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