Lesson HOW TO solve linear equations

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About longjonsilver: I have a new job in September, teaching

1.4 When you are asked to solve an equation what does this mean?

To begin to fully understand the relationship between the abstract question being asked and any real question that hides a linear equation question and the power of a simple sketch of the equation in graphical form, we need to make the leap in understanding of what "solving" means.

Eg solve y=2x-8 means "put y=0 and then find the value(s), if any, of x that satisfy that". Period!

If you have read my previous Lessons on Linear Equations, then hopefully you can begin to visualise what the straight line y=2x-8 looks like:

its gradient is +2, so is a line like /.
its y-intercept is at y=-8...so the solution to this will be a positive x-value, since the line cuts the x-axis at some point greater than zero.

So, lets solve this example...

EXAMPLE:
Solve y=2x-8.

solution:
2x-8 = 0
2x = 8
x = 4

graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+2x-8%29+

graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+2x-8%29+

This is the solution of the equation. This is also called the ROOT of the equation.

A linear equation will never have more than 1 root. It will, the majority of the time, have 1 root. A select few "special" straight lines will have no solutions. These are the lines that run parallel to the x-axis, like y=3 or y=-5. These NEVER cross the x-axis, so no solution... no root.

ANY QUESTION like y=.... that says solve, means put y equal to zero.

1.5 Explaining more tricky questions

What does "Solve 2x-2 = 4-x" really mean? Physically, it means "if we plotted the 2 straight lines y=2x-2 and y=4-x, where would they cross?". Again, we expect a maximum of just 1 answer, since 2 straight lines will only cross once, unless they are parallel, then they will never cross and hence no solution.

EXAMPLE
Solve 2x-2 = 4-x

solution:
3x-2 = 4
3x = 6
x = 2

So, these 2 straight lines cross when x=2.
graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+2x-2%2C+4-x%29+

graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+2x-2%2C+4-x%29+

The secret to understanding maths (any maths) is visualising what is being asked. So, draw a picture, sketch a graph and these will help you...with practice :-)

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