Question

OPEN-ENDED Find three values for \(m\) so the system has no solution, one solution, and two solutions. Justify your answer using a graph. $$ \begin{aligned} &3 y=-x^2+8 x-7 \\ &y=m x+3 \end{aligned} $$

Short answer

To have no solution, \(m\) should be equal to 0 with a y-intercept of 3. To have exactly one solution, \(m\) should be equal to 0 with a y-intercept greater than 7. And the system can't have infinite solutions as one equation is a parabola and the other is a line.

Step-by-step solution

Finding value for no solution

For the system to have no solution, the line must not intersect the parabola. This occurs when \(m\) is equal to the slope of the tangent to the parabola at the point of tangency. The derivative of the parabola, \(y'=-2x+8\), gives us the slope of the tangent line. Setting this equal to \(m\) and solving for \(x\) gives \(x=4\). Substituting \(x=4\) into the equation for the parabola, we find \(y=-7\). So, the slope of the tangent line at that point is \(m=-2(4)+8=0\). Therefore, the line \(y=0x+3\) or simply \(y=3\) will not intersect the parabola, yielding no solution.

Finding value for one solution

For the system to have exactly one solution, the line must be tangent to the parabola. We've already found the slope of the tangent line in Step 1. Therefore, if we shift the line \(y=0x+3\) upwards so that it becomes tangent to the parabola, we'll have exactly one solution. To do this, we need to set \(m=0\) and change the y-intercept from 3 to something greater than 7 (the vertex of the parabola). For this example, we'll use the line \(y=0x+8\). At \(x=4\), both the parabola and the line will have \(y=8\), resulting in a single solution.

Finding value for infinite solutions

For there to be infinite solutions, the given equations would both need to represent the same line, which is not possible in this system as one is a parabola and the other is a line. Therefore, there can't be a value for \(m\) that would result in infinite solutions.

Graphical Confirmation

Graph each of these scenarios to confirm. The graph will depict a parabola and: a line not intersecting the parabola for no solutions (when \(m=0\)), a line tangent to the parabola for one solution (when \(m=0\) but y-intercept is greater than 7), and no case for infinite solutions.

Key concepts

Quadratic Equations

A quadratic equation is an equation that involves a variable raised to the second power, commonly written in the standard form as \( ax^2 + bx + c = 0 \). In this exercise, we deal with the parabola represented by the quadratic equation \( 3y = -x^2 + 8x - 7 \). By rearranging, we get \( y = -\frac{1}{3}x^2 + \frac{8}{3}x - \frac{7}{3} \). This equation describes a U-shaped curve that can open upwards or downwards. Here, it opens downwards because the coefficient of \(x^2\) (the term \(-\frac{1}{3}\)) is negative.

Understanding the nature of parabolas is key in solving systems involving quadratic equations. The vertex is the highest or lowest point on the curve, and it can be found using the formula \( x = -\frac{b}{2a} \). For our parabola, this occurs at \( x = 4 \). The vertex provides critical insight, especially when determining intersections with a line.

Solutions to Systems

The exercise involves finding how a line and a parabola interact to result in different types of solutions to the system of equations. There are three scenarios:

• No Solution: This occurs when the line does not cross the parabola. In mathematical terms, this translates to choosing a value for \(m\) that makes the line parallel to the tangent of the parabola, but with a different y-intercept. In our problem, \(m=0\), with a y-intercept of 3 (i.e., \(y=3\)), results in no intersection.
• One Solution: The line is tangent to the parabola, touching it at just one point. This happens when the slope of the line \(m\) equals the slope of the parabola at that tangent point, and the line passes through the point of tangency. In the example, \(m=0\) and shifting the y-intercept to 8 makes the line tangent to the parabola at \(x=4\).
• Two Solutions: Here, the line crosses the parabola at two distinct points. Adjusting \(m\) and the y-intercept just right results in such intersections. However, specific values aren't given in the problem, because it mainly focuses on zero and one solution scenarios.
  

Graphical Analysis

Graphical analysis is a powerful method to visualize and confirm solutions to systems of equations. In this problem, we plot both the quadratic equation and the line for different values of \(m\) to observe their points of intersection.

• Graph for No Solution: A graph with the parabola and the horizontal line \(y=3\) shows no intersection, confirming no solutions.
• Graph for One Solution: When the line is \(y=8\), it touches the parabola at just one point (tangent), indicative of one solution.
• Graph for Two Solutions: Though not explicitly shown, one can infer that if we adjusted \(m\) and the y-intercept appropriately (not specified here), the line would cross the parabola twice.
  

Graphing not only assists in verifying solutions but also aids in understanding how changes in the line's slope and intercept affect the nature of its intersection with the parabola.