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Anywhere I've looked, the definition of solution of equation is root(s) of that equation. But why does a textbook say that equation $y=7-3x$ has infinite number of solutions?

Thanks

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    $\begingroup$ It's the equation of a line, which has infinitely many points. There are infinitely many pairs $(x,y)$ that solve it. $\endgroup$ Jan 30, 2019 at 19:42
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    $\begingroup$ A root corresponds to a solution with $y=0$, but a solution is a pair $(x,y)$ that satisfies the equation ($y$ doesn't have to be $0$). $\endgroup$ Jan 30, 2019 at 19:43
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    $\begingroup$ Take $(x,y)=(0,7),(1,4),(2,1),(3,-2),(4,-5),\ldots $, so we have already infinitely many integer solutions. $\endgroup$ Jan 30, 2019 at 19:47
  • $\begingroup$ You need as many equations as you have variables to find the solutions of these sorts of equations. $\endgroup$
    – John Doe
    Jan 31, 2019 at 0:28

3 Answers 3

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Equations have solutions, but not roots. Polynomials have roots, but not solutions.

  • The equation $y=7-3x$ has infinitely many solutions.
  • The equation $0=7-3x$ has only one solution.
  • The polynomial $7-3x$ has only one root, which is the solution of the equation $0=7-3x$.
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Solutions are not the same as roots.

A root is a value of $x$ where $y = 0$ and your equation is satisfied. In your equation, the root is $x = 7/3$.

A solution in general is a pair of values $(x,y)$ that satisfy the equation. Examples of solutions to your equation include, but are not limited to, $(7,-14)$, $(0,7)$, and $(\frac 73, 0)$. There are an infinite number of such pairs.

Roots are a special type of solution. But not all solutions are roots.

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This is because a root is just setting the y value equal to 0 and solving for x, whereas the equation itself is just saying "how many points are there such that y = 7 - 3x", not "how many points are there such that the y value is 0". For example, (2, 1) would be a solution to this equation, but not a root.

Hope this helps!

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