# System of Equations: Find m, Graphical Interpretation

In summary, the conversation discusses a problem that involves finding the slope of a line that intersects the graph of x = y^2 only at (4, 2). The suggested method for solving this problem is to use the quadratic formula, setting the coefficients equal to find the value of m. The conversation also mentions using Fermat's method to find the slope of the tangent line, which involves setting the quadratic equation equal to (y-2)^2 and expanding it to solve for m and b. The student is unsure of how to proceed after finding that the problem has only one solution. They also have questions about the process and properties involved.

## Homework Statement

Shown in the figure is the graph of x = y^2 and a line of slope m that passes through the point (4, 2). Find the value of m such that the line intersects the graph only at (4, 2) and interpret graphically.

x = y^2
y = mx + b

## The Attempt at a Solution

Since this is a system of two equations I went ahead and plugged x into y = mx + b. This gives me the equation y = m(y^2) + b. I then get by^2 - y + b = 0. I can plug this into the quadratic formula (or so I think). This gives me y = -(1 +- sqrt(1 - 4b^2)) / 2. I am not sure where to go from here.

I think you might be overcomplicating this one.

Graph out the function provided, then determine the slope of a line through the given point that would satisfy the conditions. The slope of your line is what is important to determining whether or not it will touch more than 1 location on the function's curve.

Do you have the "figure" that is mentioned in the instructions? That might make understanding the problem a bit easier.

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I do have the figure. It is simply a graph of x = y^2 and another line intersecting x = y^2 at (4, 2). I am given no other information on the line that is intersected x = y^2. I am instructed to find the slope of the line so that it only intersects x = y^2 at (4, 2). I have the answer to this problem, but do not know how to get it. The only hints I got from my instructor dealt with using the quadratic formula to solve for x and then for m.

Yes, you are given "other information"! You are told that the line intersects the parabola only at (4, 2)- there are no other intersections. If you look closely at your graph you should see that the only line that intersects the parabola at (4, 2) and no where else is the tangent line to the parabola there.

You could find the slope of the tangent line by differentiating the function but since this is "PreCalculus", you probably are not supposed to and you don't have to. Here is Fermat's method for finding tangents that predates Calculus:

Saying that y= mx+ b (so x= y/m- b/m) intersects $x= y^2$ means, of course, that $x= y^2= y/m- b/m$ or $y^2- (1/m)y+ b/m= 0$. That's the quadratic equation you want to solve.

And saying that the line intersects the parabola only at that point means that the quadratic equation has only one solution- in fact, you know that that one solution must be y= 2. So you must have $y^2- (1/m)y+ b/m= (y- 2)^2$. Expand the right side and set the coefficients equal to find m and b.

(Here, I am not saying you should use the quadratic formula- but you could: the quadratic equation $ax^2+ bx+ c= 0$ has a double root if and only if its "discriminant", $\sqrt{b^2- 4ac}$, is 0.)

That is a great reply. I've gotten a little further, but am stuck again. I used the Quadratic Formula to solve this and came up with y = (-1/m +- sqrt((1/m^2) - (4b - m))) / 2. I know that the problem has only one solution so sqrt((1/m^2) - (4b - m)) = 0. I am not sure what to do from this point.

Also, I am not understanding how you got y^2- (1/m)y+ b/m= (y- 2)^2. Is there a law or property concerning this? I also do not know what it means to set the coefficients equal. This is my first math course in over 6 years. I have forgotten quite a bit. I tried to look in my textbook for any hints or clues, but there are no examples that deal with this specific kind of problem. I am sure I can use the basics from other examples to deal with this, but I simply do not know how.

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## 1. What is a system of equations?

A system of equations is a set of two or more equations that have multiple variables and must be solved simultaneously in order to find the values of those variables.

## 2. How do you find the value of m in a system of equations?

To find the value of m in a system of equations, you must first isolate m on one side of the equation and solve for it using algebraic methods. This can include combining like terms, using the distributive property, and solving for the variable using inverse operations.

## 3. How is a system of equations graphically interpreted?

A system of equations can be graphically interpreted by graphing each equation on the same coordinate plane and finding the point of intersection between the two lines. This point represents the solution to the system of equations and can be used to find the value of m.

## 4. What is the significance of m in a system of equations?

m is a variable that represents the slope of a line in a system of equations. It is important because it tells us the rate at which the line is changing and can be used to make predictions and solve real-world problems.

## 5. Can a system of equations have more than one solution for m?

Yes, a system of equations can have more than one solution for m. This can happen if the two equations in the system are parallel lines, meaning they have the same slope and will never intersect. In this case, any value of m will satisfy both equations and be a solution to the system.

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