Problem 53 - College Algebra

Question

Make a rough sketch in a rectangular coordinate system of the graphs representing the equations in each system. The system, whose graphs are a line with positive slope and a parabola whose equation has a positive leading coefficient, has two solutions.

Step-by-Step Solution

Answer

The sketch will show a line rising from left to right (representing the line with a positive slope) and a 'U' shaped curve opening upwards (representing the parabola with a positive leading coefficient). The points where these two graphs intersect are the solutions to the system of equations.

Step 1: Sketching the Line with a Positive Slope

Begin by sketching a line with a positive slope. This line should ascend from the lower left to the upper right of the graph as you move along the x-axis in a positive direction. Any line of the form \(y = mx + c\) where \(m > 0\) will work.

Step 2: Sketching the Parabola with a Positive Leading Coefficient

Then, draw a parabola with a positive leading coefficient. This means the parabola opens upwards and can be represented as function of the form \(y = ax^2 + bx + c\) where \(a > 0\). Position this parabola on the graph ensuring it intersects the line from step 1 at two different points, as stated in the problem.

Step 3: Identify the Solutions

The solutions to the system of equations are the points where the line and the parabola intersect. Mark these points on the sketch.