Question

In Exercises 47–52, solve the nonlinear system. Justify your answer with a graph. $$ \begin{aligned} &y^2=4 x+17 \\ &y=x+5 \end{aligned} $$

Short answer

The solutions to the nonlinear system of equations are (-1,4) and (4,9).

Step-by-step solution

Substitute \(y\)

Substitute \(y\) from the second equation \(y=x+5\) into the first equation to get: \((x+5)^2 = 4x + 17\).

Solve for \(x\)

After expanding and rearranging the equation, you get a quadratic equation in the form of \(x^2 -3x -4=0\). Solve this equation by using factoring or applying the quadratic formula \(x = [-b ± sqrt(b^2 -4ac)] / 2a\). The solutions for \(x\) are \(x= -1\), and \(x=4\)

Find \(y\)

Substitute the values of \(x\) into the second equation to get the corresponding \(y\) values. When \(x=-1\), \(y=-1+5=4\) and when \(x=4\), \(y=4+5=9\). So, the solution to the system of equations is (-1,4) and (4,9).

Justify with a graph

Plot the graphs of both equations on the same set of axes. The points where the two graphs intersect are the solutions to the system. In this case, these intersection points will be (-1,4) and (4,9).

Key concepts

Quadratic Equations

Quadratic equations are a fundamental aspect of algebra and appear in a variety of mathematical contexts. These equations can be identified by their standard form, which is \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The solutions to a quadratic equation, also known as the roots, are the values of \(x\) that make the equation true. To solve a quadratic equation, one can use several methods, such as factoring, completing the square, or applying the quadratic formula, \( x = [-b \pm \sqrt{b^2 -4ac}] / 2a \).

In the exercise provided, the quadratic equation emerges after using the substitution method to combine two equations into one. The resulting equation, \( x^2 -3x -4=0 \), is then solved to find the values of \(x\) that satisfy both original equations. Factoring or applying the quadratic formula to this equation yields two possible solutions for \(x\), which is typical for quadratic equations unless they have a repeated root.

Substitution Method

The substitution method is a powerful technique to solve systems of equations, including nonlinear systems like the one in the exercise. This method involves expressing one variable in terms of another from one equation and substituting this expression into the other equation. By doing so, you effectively reduce the system to a single equation in one variable, which is often easier to solve.

For instance, in the provided exercise, we start with two equations: \(y^2=4x+17\) and \(y=x+5\). By expressing \(y\) as \(x+5\) from the second equation and substituting it into the first, we transform the system into \( (x+5)^2 = 4x + 17 \), a single quadratic equation. Solving this quadratic allows us to find the corresponding values of \(x\), and subsequently, by back substitution, determine the associated \(y\) values to reach the solutions of the original system.

Graphical Method

The graphical method for solving systems of equations provides a visual representation of the solutions. By plotting each equation on the same set of axes, their points of intersection represent the pairs of \(x\), \(y\) values that satisfy all the equations simultaneously. It's particularly insightful because it allows us to see the number of solutions and understand the relationship between the equations.

In the current example, plotting the parabola \(y^2=4x+17\) and the straight line \(y=x+5\) on a graph and identifying their points of intersection would reveal the solutions to the system. The graphical method not only serves as a solution technique but also as a means to verify algebraic solutions, as the intersection points \( (-1,4) \) and \( (4,9) \) found algebraically should match graphically.