Intersection of a line and a curve.

Solution 1:

Equating the two equations doesn't mean the curve and the line share a single value of $y$; it means that you're assuming they share a value of $y$, and then getting an equation for the corresponding shared value of $x$. This says nothing about how many shared pairs $(x, y)$ there might be.

Solution 2:

What is an intersection?

An intersection is where both $y$ for each function are equal for the same $x$.

Consider $f(x)=g(x)$.

Your intersection(s) are ALL the points where you can plug (the same) $x$ into both $f(x)$ and $g(x)$ and get an equality.

Because $f(x)=g(x)$, $y=y$, if that makes sense.

And this can happen say $0,1,2...\infty$ times.

Solution 3:

Define two systems of equations to be equivalent if they have precisely the same sets of solutions. Abstracting, "solving a system of equations" is the process of successively replacing a given system by equivalent systems until one reaches a "tautological" system whose solutions can be read off by inspection.

For example, the system $$ \left. \begin{aligned} y &= 2x + 3 \\ y &= x^{2} + 3x + 1 \end{aligned} \right\} \tag{1} $$ is equivalent, by subtracting the first equation from the second, to the system $$ \left. \begin{aligned} y &= 2x + 3 \\ 0 &= x^{2} + x - 2 \end{aligned} \right\} \tag{2} $$ in which $y$ has been eliminated from the second equation. You solved the second equation using the quadratic formula, obtaining $$ \left. \begin{aligned} y &= 2x + 3 \\ x &= -2\quad\text{or}\quad 1 \end{aligned} \right\} \tag{3} $$ then implicitly used the first equation to deduce the corresponding value(s) of $y$.

"Why this works" should be apparent. From this perspective, it should be clear that the reasoning

Equating the curve and the straight line means they share a single similar value of $y$ while they clearly share two.

would point to a logical gap only if the equation $y = y$ had a unique solution. But the opposite is true: $y = y$ is a tautology; it has no non-solutions.

In general, any "reversible operation" on a system of equations yields an equivalent system. The following operations (non-exhaustive list!) are reversible is this sense:

• Adding a (constant) multiple of one equation to another equation.

• Multiplying an equation by a non-zero constant, or by a non-vanishing expression.

• Exchanging two equations.

• Replacing an equation $a = b$ with $f(a) = f(b)$ for some injective function $f$. (For equations involving real variables, this includes cubing or exponentiating both sides, squaring both sides when both sides are known to be non-negative, and so forth.)

• If $f$ and $g$ are functions, replacing $f(y) = g(y)$ with $f(\phi(x)) = g(\phi(x))$ for some injective function $\phi$.

Compare the first three with the Gaussian elimination algorithm for systems of linear equations in several variables.