Testing solutions to linear equations
About this video
Sal is given a table of points (4, -8), (7, -20), (8, -24), (9, -28) and three candidate equations. He substitutes x values into each equation to rule out y = -2x + 1 and y = -2x, and shows y = -4x + 8 produces every point in the table.
You'll be able to
- test whether an ordered pair is a solution of an equation by substituting the x value.
- rule out an equation when even one point does not fit.
- explain why two points are enough to pin down a line.
Key words
- ordered pair
- A pair of numbers (x, y) that gives an input and its matching output.
- linear function
- A rule whose graph is a straight line, such as y = -4x + 8.
- substitute
- Replace a variable with a number to see what the equation gives.
Transcript
Some ordered pairs for a linear function of x are given in the table below. Which equation was used to generate this table? So they give us a bunch of data points. When x is 4, y is negative 8. When x is 7, y is negative 20, so on and so forth. And these points have been generated by one of these equations right over here, so let's just see which of these equations actually could have generated all of these points.
So let's go to this first point right over here, 4 comma negative 8, and let's go equation by equation. If x is 4 here, negative 2 times 4 would be negative 8. Let me just write it down. We'd multiply negative 2 times 4 to get negative 8, and then I would add 1 to get negative 7.
I would get negative 7. So for this equation, when x is 4, y would be negative 7, not negative 8. So this equation definitely did not generate this top point right over here. So we could just rule it out. Whatever equation should have been able to generate for any given x, for any of these x's, should generate this y right over here.
Now let's go to this next equation. y is equal to negative 2x plus 0. So when x is 4, y would be negative 2 times 4, which is negative 8. So this second equation seems to be capable of generating this first set of points. When x is 4, y is negative 8. But let's see if it works for this second one right over here.
When x is 7, you would have negative 2 times 7, which is negative 14, but here, when x is 7, y is negative 20. So this one, we can rule out, because when x is 7, y does not equal negative 20 like this point right over here. So we'll rule that second one out. Now let's look at this third one.
y is equal to negative 4x plus 8. Well, we could go to this first point again. When x is 4, let's think about what happens when x is 4. So when x is 4, you have negative 4 times 4, which is negative 16, plus 8, which is negative 8. So it seems to be able to generate this first point.
When x is 4, y is negative 8 for this equation right over here. Now let's see what happens when x is 7. So negative 4 times 7 is negative 28. Negative 28 plus 8 is negative 20, so this candidate is starting to look pretty good. It's pretty good. It satisfies these two points, and frankly, for any of these linear functions, if it satisfies any of these two points, if some linear function generated all of these, if some function can generate any two of them, it will satisfy all of these, because two points define a line.
But we can verify it right over here. So let's see what happens when x equals 8. Negative 4 times 8 is negative 32, plus 8 is negative 24. So that satisfies so that when x is 8, y does equal negative 24. And then finally, let's look at that last point. When x is 9, negative 4 times 9 is negative 36, plus 8 is equal to negative 28.
And that's exactly what we see here in this table. So we don't even have to look any further. Choice 3 satisfies the conditions that when x is any one of these things, the corresponding y that's defined by this equation is going to be when x is 4, y is negative 8.
When x is 7, y is negative 20, so on and so forth.
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