IB Mathematics AA SL equation of a straight line SL Study Notes

Equation of a Straight Line

The equation of a straight line describes a linear relationship between the variables \(x\) and \(y\). The most common forms of this equation are the slope-intercept form and the point-slope form.

1. Slope-Intercept Form

The slope-intercept form of a line’s equation is given by:

\( y = mx + c \)

• \(m\): The slope of the line, representing the rate of change of \(y\) with respect to \(x\).
• \(c\): The y-intercept, where the line crosses the y-axis (\(x = 0\)).

Example 1: Finding the Equation

Find the equation of a line with a slope \(m = 2\) and a y-intercept \(c = -3\).

Using the formula \( y = mx + c \):

\( y = 2x – 3 \)

This is the equation of the line with slope 2 and y-intercept -3.

2. Point-Slope Form

The point-slope form of a line’s equation is given by:

\( y – y_1 = m(x – x_1) \)

• \(m\): The slope of the line.
• \((x_1, y_1)\): A point on the line.

Example 2: Finding the Equation Using Point-Slope Form

Find the equation of a line passing through the point \((3, 4)\) with a slope \(m = -1\).

Using the formula \( y – y_1 = m(x – x_1) \):

\( y – 4 = -1(x – 3) \)

Simplifying this equation:

\( y – 4 = -x + 3 \)

\( y = -x + 7 \)

This is the equation of the line in slope-intercept form.

3. General Form

The general form of a line’s equation is given by:

\( Ax + By + C = 0 \)

• \(A\), \(B\), \(C\): Constants, where \(A\) and \(B\) are not both zero.

Example 3: Converting to General Form

Convert the equation \( y = 2x + 5 \) to general form.

Rearranging terms:

\( 2x – y + 5 = 0 \)

So, the general form of the equation is \( 2x – y + 5 = 0 \).

4. Finding Slope from Two Points

If two points on a line are given as \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) can be calculated using:

\( m = \frac{y_2 – y_1}{x_2 – x_1} \)

Example 4: Calculating Slope

Find the slope of the line passing through the points \((1, 2)\) and \((4, 6)\).

Using the formula \( m = \frac{y_2 – y_1}{x_2 – x_1} \):

\( m = \frac{6 – 2}{4 – 1} = \frac{4}{3} \)

So, the slope of the line is \(\frac{4}{3}\).

Key Takeaways

• Slope-Intercept Form: Best for quickly identifying the slope and y-intercept of the line.
• Point-Slope Form: Useful when you know a point on the line and the slope.
• General Form: A standard representation that is useful in various mathematical applications.
• Finding Slope: Calculating the slope using two points helps in forming the line equation.