Question

THOUGHT PROVOKING Write a nonlinear system that has two different solutions with the same \(y\)-coordinate. Sketch a graph of your system. Then solve the system.

Short answer

The chosen nonlinear equations are \(y = x^2\) and \(y = 4 - (x - 1)^2\). These intersect at two points which lie on the same horizontal line, verifying that they have the same y-coordinate. After solving the system, we find that the solutions agree with the graph.

Step-by-step solution

Construct Equations

Choose a suitable pair of nonlinear equations that can intersect at two different points with the same y-coordinate. In this case, \(y = x^2\) and \(y = 4 - (x - 1)^2\) have been chosen.

Graph Equations

Graph the functions \(y = x^2\) and \(y = 4 - (x - 1)^2\) to visually confirm that they intersect at two different points with the same y-coordinate. A graph can be created utilizing graphing software or illustrating traditionally if graphing tools are not accessible.

Solve the System

To solve the system, set the equations equal to each other and solve for \(x\). \(x^2 = 4 - (x - 1)^2\). Expand and simplify to form a quadratic equation which can then be solved to find the two solutions using the quadratic formula.

Find Corresponding Y-values

Once the solutions for \(x\) have been calculated, substitute them into either of the original equations to find the corresponding y-coordinate. Because it is specified that both solutions have the same \(y\)-coordinate, it should confirm that regardless of which equation the \(x\) values are substituted into, the \(y\)-coordinate will be the same.

Key concepts

Graphing Nonlinear Equations

Understanding how to graph nonlinear equations is crucial for solving many mathematical problems. Nonlinear equations do not form straight lines when graphed; instead, they can take various forms like curves, parabolas, or circles. A classic example is the quadratic equation, which forms a parabola.

The process begins by plotting points that satisfy the equation, and then connecting these points to reveal the curve. In our exercise, we graphed the parabola represented by the quadratic function \(y = x^2\) and the downward opening parabola \(y = 4 - (x - 1)^2\). When graphing by hand, it's helpful to determine points at key intervals and where the equation crosses the axes. Using graphing software can make this process easier, providing a more accurate graph quickly.

Solving Systems of Equations

Solving systems of equations involves finding the values that satisfy all equations in the system simultaneously. When dealing with nonlinear systems, like in our exercise, the solutions often require methods beyond simple algebraic manipulation.

This can include substitution, where we solve one equation for a variable and substitute into another, or elimination, where we manipulate the equations to cancel out one of the variables. However, in many cases, especially with quadratic equations, we may need to employ the quadratic formula or factoring techniques to find the solutions. Graphical interpretations also help to visualize where the solutions lie by identifying points of intersection on the graphs.

Quadratic Equations

Quadratic equations are a specific type of nonlinear equation, characterized by the highest exponent of the variable being two. They take the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).

The solutions to a quadratic equation can be found algebraically through methods such as factoring, completing the square, or most commonly, using the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). The nature of a quadratic equation means it can have two real solutions, one real solution (a perfect square), or no real solutions depending on the value of the discriminant \(b^2 - 4ac\).

Intersecting Graphs

Intersecting graphs occur when two or more graphs have points in common. The solution to a system of equations graphically is represented by the intersection points of the equations' graphs. In the context of our exercise, the two parabolas intersect at points with the same \(y\)-coordinate, revealing the solutions to the system.

Identifying the intersections can often be done visually, by observing the points where the graphs cross. For accuracy, especially with complicated equations, algebraic methods are used to find the exact coordinates of the intersection points. Knowing how to interpret these intersections is instrumental when solving nonlinear systems, as it provides a visual confirmation of the solutions derived algebraically.