### Video Transcript

At which values of π₯ does the graph of the equation π¦ equals five times π₯ minus one times π₯ plus seven cross the π₯-axis?

We begin by recalling that the π₯-axis has an equation. We say itβs the line π¦ equals zero. And this means we can find the points where our graph crosses the π₯-axis by setting π¦ equal to zero and solving for π₯. This is sometimes called finding the roots of the equation. In this case, we say zero is equal to five times π₯ minus one times π₯ plus seven. And how do we solve for π₯? Well, the first thing youβre going to do is look to get rid of this five. The five is multiplying the expression π₯ minus one times π₯ plus seven. So, weβre going to divide both sides by five.

When we do, we find that zero is equal to π₯ minus one times π₯ plus seven. Now, the expressions π₯ minus one and π₯ plus seven are multiplying one another, and when they do, we get zero. And so, for the expression π₯ minus one times π₯ plus seven to be equal to zero, that means either π₯ minus one itself must be equal zero or π₯ plus seven must be equal to zero. Weβll solve the equation π₯ minus one equals zero for π₯ by adding one to both sides. So, π₯ is equal to one.

And weβll solve the equation π₯ plus seven equals zero by subtracting seven from both sides to give us π₯ equals negative seven. And so, we have two solutions for π₯. For the equation zero equals five times π₯ minus one times π₯ plus seven, π₯ can either be equal to one or it can be equal to negative seven.

Now, of course, before assuming that these are the values weβre interested in, weβre going to check what weβve done is correct. Letβs take the solution π₯ equals one. We substitute it into the expression five times π₯ minus one times π₯ plus seven. So, we get five times one minus one times one plus seven. Thatβs five times zero times eight, which is indeed zero, as we expected.

Weβll repeat this for the other value of π₯, π₯ equals negative seven. Thatβs five times negative seven minus one times negative seven plus seven. Thatβs five times negative eight times zero, which is again zero as required. And so, by solving the equation five times π₯ minus one times π₯ plus seven equals zero, we found the values of π₯ for which the graph of the equation π¦ equals five times π₯ minus one times π₯ plus seven crosses the π₯-axis to be π₯ equals one and π₯ equals negative seven.