Chapter Contents

• Systems of Equations
• 1. Simultaneous Linear Equations
• 2. Graphs of Linear Functions
• 3. Graphical Solution of Linear Systems
• 4. Algebraic Solutions of Linear Systems
• 5. Graphical Solution of non-linear Systems
• 6. Algebraic Solution of non-Linear Systems
• Systems of Equations Problem Solver

• Comments, Questions?

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2. Graphs of Linear Functions

It is very important for many math topics to know how to quickly sketch straight lines. When we use math to model real-world problems, it is worthwhile to have a sense of how straight lines "work" and what they look like.

We met this topic before in The Straight Line. The following section serves as a reminder for you.

a. Slope-Intercept Form of a Straight Line: y = mx + c

If the slope (also known as gradient) of a line is m, and the y-intercept is c, then the equation of the line is written:

y = mx + c

Example 1

The line y = 2x + 6 has slope m = 2 and y-intercept c = 6.

b. Intercept Form of a Straight Line: ax + by = c

Often a straight line is written in the form ax + by = c. One way we can sketch this is by finding the x- and y-intercepts and then joining those intercepts.

Example 2

Sketch the line 3x + 2y = 6.

Answer

Slope of a Line

The slope (or gradient) of a straight line is given by:

We can also write the slope of the straight line passing through the points (x₁, y₁) and (x₂, y₂) as:

Using this expression for slope, we can derive the following.

c. Point-slope Form of a Straight Line: y − y₁ = m(x − x₁)

If a line passes through the point (x₁, y₁) and has slope m, then the equation of the line is given by:

y − y₁ = m(x − x₁)

Example 3

Find the equation of the line with slope −3, and which passes through (2, −4).

Answer