Question

find the x and y-intercepts of the line that is parallel to y=2x+3 and passes through (2,1)

Answer 1

To find the x and y-intercepts of a line that is parallel to \( y = 2x + 3 \) and passes through the point \( (2, 1) \), we'll first determine the equation of the line.

1. **Identifying the slope**: The slope of the given line \( y = 2x + 3 \) is \( 2 \). Since parallel lines have the same slope, the slope of our desired line will also be \( 2 \).

2. **Using the point-slope form**: We can use the point-slope form of the equation of a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope.

Plugging in the slope \( m = 2 \) and the point \( (2, 1) \):
\[
y - 1 = 2(x - 2)
\]

3. **Simplifying the equation**:
\[
y - 1 = 2x - 4
\]
\[
y = 2x - 3
\]

Now we have the equation of the line: \( y = 2x - 3 \).

4. **Finding the y-intercept**: The y-intercept occurs when \( x = 0 \):
\[
y = 2(0) - 3 = -3
\]
So the y-intercept is \( (0, -3) \).

5. **Finding the x-intercept**: The x-intercept occurs when \( y = 0 \):
\[
0 = 2x - 3
\]
\[
2x = 3
\]
\[
x = \frac{3}{2}
\]
So the x-intercept is \( \left( \frac{3}{2}, 0 \right) \).

In summary, the intercepts for the line are:
- **y-intercept**: \( (0, -3) \)
- **x-intercept**: \( \left( \frac{3}{2}, 0 \right) \)