Question
the line y=kx+b passes through the points a(-1 3) and b(2 -1). Write the equation of the line.
Answers
GPT-4o mini
To find the equation of the line \( y = kx + b \) that passes through the points \( a(-1, 3) \) and \( b(2, -1) \), we can follow these steps:
1. **Calculate the slope (k)**:
The slope \( k \) can be calculated using the formula:
\[
k = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \( (x_1, y_1) = (-1, 3) \) and \( (x_2, y_2) = (2, -1) \).
Substituting the values:
\[
k = \frac{-1 - 3}{2 - (-1)} = \frac{-4}{2 + 1} = \frac{-4}{3}
\]
2. **Use the slope to find the y-intercept (b)**:
We can use the point-slope form or plug one of the points into the equation to find \( b \). Using the point \( a(-1, 3) \):
\[
y = kx + b
\]
Substitute \( x = -1 \) and \( y = 3 \):
\[
3 = \left(-\frac{4}{3}\right)(-1) + b
\]
This simplifies to:
\[
3 = \frac{4}{3} + b
\]
To solve for \( b \), subtract \( \frac{4}{3} \) from both sides:
\[
b = 3 - \frac{4}{3} = \frac{9}{3} - \frac{4}{3} = \frac{5}{3}
\]
3. **Write the final equation**:
Now that we have \( k = -\frac{4}{3} \) and \( b = \frac{5}{3} \), we can write the equation of the line:
\[
y = -\frac{4}{3}x + \frac{5}{3}
\]
And that is the equation of the line passing through the points \( a(-1, 3) \) and \( b(2, -1) \).
1. **Calculate the slope (k)**:
The slope \( k \) can be calculated using the formula:
\[
k = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \( (x_1, y_1) = (-1, 3) \) and \( (x_2, y_2) = (2, -1) \).
Substituting the values:
\[
k = \frac{-1 - 3}{2 - (-1)} = \frac{-4}{2 + 1} = \frac{-4}{3}
\]
2. **Use the slope to find the y-intercept (b)**:
We can use the point-slope form or plug one of the points into the equation to find \( b \). Using the point \( a(-1, 3) \):
\[
y = kx + b
\]
Substitute \( x = -1 \) and \( y = 3 \):
\[
3 = \left(-\frac{4}{3}\right)(-1) + b
\]
This simplifies to:
\[
3 = \frac{4}{3} + b
\]
To solve for \( b \), subtract \( \frac{4}{3} \) from both sides:
\[
b = 3 - \frac{4}{3} = \frac{9}{3} - \frac{4}{3} = \frac{5}{3}
\]
3. **Write the final equation**:
Now that we have \( k = -\frac{4}{3} \) and \( b = \frac{5}{3} \), we can write the equation of the line:
\[
y = -\frac{4}{3}x + \frac{5}{3}
\]
And that is the equation of the line passing through the points \( a(-1, 3) \) and \( b(2, -1) \).