Question

True or false. it is possible for a system of equations that has one linear equation and one quadratic equation to have the following number of solutions.

Answer 1

It is true that a system with one linear and one quadratic equation can have zero, one, or two solutions, depending on how the straight line intersects with the parabolic curve on a graph.

Step-by-step explanation:

It is true that a system of equations that includes one linear equation and one quadratic equation can have zero, one, or two solutions. A linear equation is represented as y = mx + b and has a straight-line graph, whereas a quadratic equation is usually in the form y = ax² + bx + c and has a parabolic shape. When these two are plotted on a two-dimensional graph, they may intersect at two points, one point, or not at all, depending on the specific equations involved.

The reason behind the different numbers of possible intersections lies in the shape of the graphs. The line can either cross the parabola twice, touch the parabola only once (at the vertex), or not intersect the parabola at all, if the line lies completely above or below the parabola. Therefore, depending on the relative positions of the line and the parabola, multiple solutions might be obtained. For real-life situations modeled by these equations, often only positive, real solutions have significance.

Answer 2

Answer: A linear equation and a quadratic equation will have either zero, one, or two points of intersection. Each point of intersection is a solution to the system of equations. In this case, we see that the graphs for these equations intersect exactly two times, at (−2,3) and (1,6).