Problem 53 - Algebra and Trigonometry

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 sketches of the line \(y = x\) and the parabola \(y = x^2\) intersect at two points (0,0) and (1,1). Hence, the system has two solutions which are these points of intersection.

Step 1: Sketching the Line

Start by drawing a line with a positive slope on the coordinate system. Recall that a positive slope indicates the line rises from left to right. For example, consider the line with equation \( y = x \) . This is a simple line with positive slope that passes through the origin.

Step 2: Sketching the Parabola

Next, sketch a parabola with a positive leading coefficient. The positive leading coefficient means the parabola opens upwards. Consider, for instance, the parabola with equation \( y = x^2 \). This parabola also passes through the origin and opens upwards.

Step 3: Identify the Solutions

The solutions to the system are where the line and the parabola intersect. For the equations \( y = x \) and \( y = x^2 \), we can see that they intersect at two points: (0,0) and (1,1). These are the two solutions to the system.