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Chapter 5: Problem 53

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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