Graph Intersections as Simultaneous Solutions

A line and a curve are drawn on one grid, and they cross in two places. The question says: use the graph to solve y equals x plus one and y equals x squared minus one. Nothing on that grid looks like a solution. Where are the answers, and why would a picture hold them at all?

This is video six of eight in Quadratic and Simultaneous Equations, and none of it needs solving by algebra. Two crossing straight lines turn up on Foundation papers too. A line crossing a curve, like this one, is Higher tier.

To see why, start with the line on its own, y equals x plus one. Every point on it is a pair of numbers, an x and a y, where the y is the x plus one. Take the point two, three. Two plus one is three, so the equation holds, and the point sits on the line. Now take two, five. Two plus one is not five, so the equation fails, and that point sits above the line. A graph is a picture of every pair that makes its equation true. On the line, the equation holds. Off the line, it fails. Now test it. Which is on y equals x plus one: three, four; four, three; or one, one? Three, four. Three plus one is four, so it sits on the line. Four, three misses, because four plus one is five, and one, one misses, because one plus one is two. That test is what the rest of this video stands on: a point on a graph makes that graph's equation true.

Now draw a second graph on the same grid, the curve y equals x squared minus one. The same test works for it: every point on the curve makes its own equation true. The line and the curve cross in two places. A place where two graphs cross is called a point of intersection. Reading off the grid, one is at two, three, and the other is at minus one, zero. Why must a point of intersection of the line and the curve make both equations true? Because it sits on both graphs. It is on the line, so it makes the line's equation true. It is on the curve as well, so it makes the curve's equation true. Simultaneous means at the same time. A pair of values that makes both equations true at the same time is a solution to the simultaneous equations. That is why each point of intersection is a solution. Check it with numbers, starting with two, three, where x is two and y is three. In the line's equation, x plus one is two plus one, which is three. That matches y, so it holds. Now the curve's equation. Two squared is four, and four minus one is three. That matches y again, so the same pair holds in both. What does x squared minus one come to when x is minus one? Zero. Minus one squared is one, and one minus one is zero, which matches y. It fits the line too, because minus one plus one is zero. Think of a crossroads, which is also called an intersection. Stand in the middle of it and you are on both roads at once. A point of intersection is that spot on two graphs. Your handle for this video: the crossroads is on both roads. A point of intersection is on both graphs, so it solves both equations.

Now the answer line. Each point of intersection gives one solution. For these two graphs you write x equals two, y equals three, and then x equals minus one, y equals zero. Two points of intersection, two solution pairs. Here is an answer that has gone wrong: x equals two, or x equals minus one, and nothing more. What is missing from that answer, and why does it matter? The y values. A solution is a whole pair, so x equals two is only half of a solution. Each x needs the y from its own point of intersection. Here's a different grid, with a straight line and a curve drawn on it. They cross at two, five and at minus one, two. What are the two solutions? x equals two, y equals five, and x equals minus one, y equals two. Each pair comes off one point of intersection, with no working. Not every point of intersection lands on the grid lines. The line y equals two x meets the same curve, y equals x squared minus one, and its upper crossing falls between the grid lines. Read each coordinate to the nearest small square and write it as an estimate: x is about two point four, and y is about four point eight. A drawn graph is only as accurate as your reading of its grid. One examiner's report says this. Some attempts indicated a lack of understanding of the solution of two equations being the point of intersection. Some solved the equations using algebraic methods. In plain words, some students did not see that the solutions are where the two graphs cross. Others solved with algebra instead, which is the long way round when the crossing points are already drawn. The fix is one habit. When two graphs are drawn for you, find where they cross first, and write each point of intersection as an x with its own y.

Let's finish with three questions, and not one of them needs any working. What does a point on a graph tell you about that graph's equation? Its coordinates make the equation true. That is what being on the graph means. Next one. Two graphs cross at one, three and at four, nine. What are the solutions? x equals one, y equals three, and x equals four, y equals nine, read straight off the crossings with no working. One more. Your handle for this video: the crossroads is on what? Both roads. Like the crossroads, a point of intersection is on both graphs, so its coordinates make both equations true at once. And that answers the question on the grid at the start. Use the graph to solve y equals x plus one and y equals x squared minus one: x equals two, y equals three, and x equals minus one, y equals zero.

If you could teach this one to a friend, give it a thumbs-up and it is done. If not, save the playlist and sketch two crossing graphs on scrap paper once, because the picture tends to stick.

Next in the chapter: Solving Equations by Iteration.

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