# Why does my textbook say that equation $y = 7-3x$ has infinite number of solutions while it has only one root?

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

• It's the equation of a line, which has infinitely many points. There are infinitely many pairs $(x,y)$ that solve it. – Lord Shark the Unknown Jan 30 at 19:42
• 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 $0$). – Michael Burr Jan 30 at 19:43
• Take $(x,y)=(0,7),(1,4),(2,1),(3,-2),(4,-5),\ldots$, so we have already infinitely many integer solutions. – Dietrich Burde Jan 30 at 19:47
• You need as many equations as you have variables to find the solutions of these sorts of equations. – John Doe Jan 31 at 0:28

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 $$0=7-3x$$.

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.

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!