How do you determine whether a linear system has one solution, many solutions, or no solution when given 2x-5y=3 and -4x+10y= -6?

Linear `->` straight line plot

When in the form of `y=mx+c` and you compare them.

`m ->` gradient (slope)

`c ->` y-intercept (point where it crosses the y-axis)

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With two linear equations

If `m` is the same in each but `c` is not the same then
`color(white)("dddd")`Parallel so do not cross thus no shared point.
`color(white)("dddd")`This is called 'no solution'.

If both `m and c` are the same
`color(white)("dddd")`One is superimposed on the other (coincidental).
`color(white)("dddd")`This is called an' infinite count of solutions'.

If both `m and c` are different then they cross once.
`color(white)("dddd")`This has just one shared point so has 1 solution.
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Given equations:

`color(white)(d.)2x-5y=3" ".....................Equation(1)`
`-4x+10y=-6" "................Equation(2)`

Manipulation gives:

`y=2/5x-3/5" ".....................Equation(1_a)`

`y=2/5x-3/5" "........................Equation(2_a)`

Both `m and c` are the same so they are coincidental.
Thus there is an infinite count if solutions
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`color(blue)("Check")`

`(-2)xxEquation(1) -> -4x+10y=-6`

This is the same as `Equation(2)`