Section 2.3: Writing Linear Equations
There are several forms to write a linear equation:
- Slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Point-slope form: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line.
- Standard form: \( Ax + By = C \), where A, B, C are integers and A ≥ 0.
Choosing the form depends on the information given: slope & intercept, slope & point, or two points.
Key Tips:
- Use point-slope form when you know a point and the slope.
- Convert to slope-intercept form for easy graphing.
- Standard form is often preferred for algebraic manipulation.
Example 1
Write the equation of a line with slope \( m = 3 \) and y-intercept \( b = -2 \).
Slope-intercept form: \( y = 3x - 2 \)
Example 2
Write the equation of a line passing through point (2,5) with slope \( -1 \).
Use point-slope form: \( y - 5 = -1(x - 2) \) → \( y - 5 = -x + 2 \) → \( y = -x + 7 \)
Example 3
Write the equation of the line passing through points (1,2) and (3,6) in standard form.
Slope: \( m = \frac{6-2}{3-1} = 2 \) Slope-intercept form: \( y - 2 = 2(x - 1) \) → \( y = 2x \) Standard form: \( 2x - y = 0 \)
Practice Problems
- Write the equation of a line with slope 4 and y-intercept 1.
- Find the equation passing through (0,-3) with slope 2.
- Write the equation of a line through points (2,4) and (5,10) in slope-intercept form.
- Convert \( y - 3 = 0.5(x - 2) \) to standard form.
- Write the equation of a line with slope -3 passing through (1,1) in slope-intercept form.