Question 38·Medium·Linear Equations in One Variable

For a linear equation in $x$ to have infinitely many solutions, what must be true about the two expressions on each side of the equation? Should they intersect once, never, or be exactly the same line?

Once you have both sides written as a constant times $(x - 5)$, focus on those constants. What must be true about them for the two expressions to be identical for all $x$?

In one line, type y = m(x - 5) and let Desmos create a slider for m. In another line, type y = 7x - 35.

Adjust the slider for m and watch the two lines. You are looking for the value of m where the two lines lie exactly on top of each other (they become a single line instead of two distinct ones).

When the lines fully overlap for all visible $x$-values, note the corresponding value of m on the slider. That is the value that makes the original equation true for every $x$ (infinitely many solutions).

The left side is already in factored form: $m(x - 5)$.

Factor $7x - 35$ on the right side:

So the equation becomes

A linear equation has infinitely many solutions only if both sides are the same expression for every $x$.

In the rewritten equation

this will happen only if the two multiples of $(x - 5)$ are equal, so the entire left side and right side match for any $x$.

Since both sides are the same factor $(x - 5)$ multiplied by different constants, those constants must be equal:

With this value, the equation becomes $7(x - 5) = 7(x - 5)$, which is true for all $x$, so the equation has infinitely many solutions when $m = 7$.