Use graphs, not formulas. For instance, the secant method is a sort of linearization method, and you can explain how to find the solution (the zero) just by drawing graphs.

Use examples from the fields of expertise of the students.

Geometry is much more convincing than algebra.

As explained by Herbert geometry would be more helpful than algebra. For instant you may demonstrate to non-mathematics students [by the way of plotting various points on a graph paper and joining them] that a set of (x, y) couplets join into a straight line that passes through (0, 0) only if the ratio y/x is same [say a constant = m] for all the (x, y) this can algebraically be written as y/x = m or y = mx. If the ratio varies then the points do not form a straight line. This ratio is what we call the slope.

Further, demonstrate to the students that a line would intersect y-axis only at one point, and if it is other than (0, 0) say (0, c) then you can show that all the points (x, y) falling on a common line would have the relationship y = mx + c.

Hope it helps.

For students who don't have mathematics background "Painless Algebra" by Lynette Long is very good textbook.

CAS supported real world applications

By using maple or mathematica

Or hand-held

Are you familiar with the Hand-on Equations (R) materials?

Take a look at how this 8 years-old boy solves an equation using the balance model: https://www.youtube.com/watch?v=HBf25kNs0tA.

What I do with the students I teach (who are training to be primary school mathematics teachers) is to get them to create and build patterns based on information provided. For instance input 1 and output 3; input 2 and output 5; input 3 and output 7 etc. They then have to build the next few steps - e.g. input 4 output ? or output 13 input ?. They then determine the output for any input (x) and an input for a given output (y) and write in words the relationship between input and output. In this instance the relationship is multiply the step number by 2 and add one to determine the output. They can then write this in symbols as y (output) = input (x) time 2 add one. It is then a small step to understand y = mx+ b as output = input of x times m add the value of b.

Once I have done this a few times I then use the Hands On Equations materials suggested by Helia. By this stage they can see that the linear equation describes a relationship between input and output governed by the function. My experience has been that for many students, this is their first experience of understanding that algebra isn't about the symbols, but rather it uses symbols to express relationships. Only later do I start to graph the relationship to strengthen the understanding.