SOLUTION: Can you please explain to me how a linear equation can have 2 solutions? Thanks.

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Question 1123589: Can you please explain to me how a linear equation can have 2 solutions? Thanks.
Found 4 solutions by greenestamps, MathLover1, solver91311, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!

A linear equation is an equation whose graph is a straight line.

Each point on the graph is a solution to the equation.

A line consists of an infinite number of points; so a linear equation has not only 2 solutions, but an infinite number of solutions.

Example: y = 2x (a very simple linear equation)

Choose any value for x; the corresponding y value is twice the x value. Clearly you have an infinite number of choices for the x value.

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And having written that response, I'm wondering if you meant a linear equation in one variable, like 3x+2=11.

If that is what you mean, then it is NOT possible for a linear equation to have two solutions. A linear equation in one variable always has a single solution.

Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website!

a linear equation can have solutions, solution or solutions
go to:
https://youtu.be/qsL_5Y8uWPU

Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!

In order to make sense of your question, we need to establish precisely what you mean by "linear equation". Linear equations in one variable have exactly one element in the solution set. Linear equations in more than one variable have infinite elements in their solution set. If you are including a single-variable equation where the variable is contained in absolute value bars then you could have a solution set with two elements. However, it would be a stretch to call such a thing a linear equation since it would actually be the composite of two linear relationships.

Please share an equation that you or anyone else asserts is both linear and has exactly two solutions.

John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52781)   (Show Source):
You can put this solution on YOUR website!
.

The standard form of a linear equation in one unknown is ax = b, (1) where "a" and "b" are numbers (given constant values), x is unknown. You may have these 3 and only 3 cases. a) a =/= 0. Then the equation (1) has a single solution x = . Geometrically, it corresponds to the case when the sloped straight line y = ax intersects the horizontal line y = b at a unique point. b) a = 0. Then you may have two sub-cases: b1) b = 0. Then you have INFINITELY many solutions for the equation (1): any value of x satisfies the equation 0*x = 0. In particular, two values x= 1 and x=2 are the solutions, and it answers the problem's question. Geometrically, it corresponds to the case when the horizontal straight line y = 0 = 0*x coincides with the coordinate x-axis. b2) b =/= 0. Then the equation (1) has no solution. Indeed, in this case the left side of the equation (1) 0*x is identically equal to zero, while the right side is not zero. Geometrically, it corresponds to the case when the horizontal straight line y = 0 = 0*x is parallel and does not coincide with the horizontal line y = b.

The lesson to learn from this solution is THIS :

If a linear equation ax = b has two solutions, then it has infinitely many solutions.

Or in this EQUIVALENT form :

A linear equation ax = b has two solutions if and only if it has infinitely many solutions.

Corollary :

If a linear equation ax = b has two solutions, then it is DEGENERATED, i.e. has the leading coefficient zero (ZERO) at x.