BeginnerTopic 7 of 104Engineering Mathematics
Linear equations
Equations of the form ax+b=0 whose graphs are straight lines; the base case of all modelling.
In depth
A linear equation ax+b=0 has the single solution x=-b/a. In two variables y=mx+c is a straight line with slope m and intercept c. Systems of linear equations are solved by substitution, elimination or matrices.
Key points
- Slope = rise/run
- Two lines meet at one point unless parallel or identical
- Linear models are the first approximation to many systems
ExampleA spring with k=200 N/m has force F=200x, a line through the origin.
Linear models describe proportional relationships, such as Hooke's law or Ohm's law, and they are the first approximation used for almost any system.
What is Linear equations?
Equations of the form ax+b=0 whose graphs are straight lines; the base case of all modelling.
Why is it important to understand linear equations?
Linear models describe proportional relationships, such as Hooke's law or Ohm's law, and they are the first approximation used for almost any system.
What does the slope of a linear relationship mean physically?
It is the rate of change, such as stiffness in N/m or resistance in ohms, and the intercept is the value when the input is zero.
When is a linear model not good enough?
When the response bends, saturates or has a threshold, such as a diode or a yielding material. Then a nonlinear model is needed.
How is Linear equations used in practice?
Typical applications include modelling a spring force with F=kx, converting a thermocouple voltage to temperature over a small range and estimating cost as a fixed charge plus a rate per unit.
What should I learn before Linear equations?
Study "Algebraic manipulation" first (topic 6 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Linear equations?
Slope = rise/run. Two lines meet at one point unless parallel or identical. Linear models are the first approximation to many systems.
Can you give an example of Linear equations?
A spring with k=200 N/m has force F=200x, a line through the origin.
What comes after Linear equations?
The next topic is "Quadratic equations", which builds directly on this one.
How difficult is Linear equations?
It is classed as Beginner in Engineering Mathematics (topic 7 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.