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?

1 Answer
Jun 11, 2018

Infinite count of solutions

Explanation:

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)