Problem 13 - Precalculus with Limits

Question

Solving a System by Substitution In Exercises \(7-14,\) solve the system by the method of substitution. Check your solution(s) graphically. $$\left\\{\begin{array}{r}{x^{2}+y=0} \\ {x^{2}-4 x-y=0}\end{array}\right.$$

Step-by-Step Solution

Answer

The solutions to the system of equations are \((0, 0)\) and \((2, -4)\).

Step 1: Isolate \(y\) in one equation

Take the first equation \(x^{2}+y=0\). Isolate \(y\) to obtain \(y=-x^{2}\).

Step 2: Substitute into the second equation

Replace \(y\) in the second equation \(x^{2}-4 x-y=0\) with \(-x^{2}\) from Step 1 to get the equation \(x^{2} - 4x - (-x^{2}) = 0\). This simplifies to \(2x^{2} - 4x = 0\).

Step 3: Solve for \(x\)

Factor the equation \(2x^{2} - 4x = 0\) to obtain \(2x(x - 2) = 0\). This gives the roots \(x = 0\) and \(x = 2\).

Step 4: Solve for \(y\)

Substitute \(x = 0\) and \(x = 2\) into \(y=-x^{2}\) (from Step 1) to find the corresponding \(y\) values. Thus, when \(x = 0\), \(y = -(0)^2 = 0\), and when \(x = 2\), \(y = -(2)^2 = -4\). Therefore, the solutions to the system are \((0,0)\) and \((2,-4)\).

Step 5: Check solutions graphically

Plot the two equations on the same graph and check whether the points \((0, 0)\) and \((2, -4)\) lie at the intersection of the two curves. If they do, this validates the solution.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations by replacing one variable with an expression involving the other variable. This method streamlines finding a solution by reducing a two-variable system into a single variable equation.

For example, when you are given a system such as: \[\left\{\begin{array}{r}{x^{2}+y=0} \{x^{2}-4x-y=0}\end{array}\right.\], you first isolate one of the variables in one of the equations, like isolating \(y\) in the first equation to get \(y=-x^{2}\). Then, you substitute this expression for \(y\) into the other equation. You continue simplifying and solving until you find the values for the original variables. It's best to start with an equation where a variable is already on its own or can be easily isolated.

Systems of Equations

Systems of equations involve finding values for variables that make all of the equations in the system true simultaneously. They come in many forms, including linear, quadratic, and higher-degree systems.

A system with at least one quadratic equation, like \(x^{2}+y=0\), is considered a nonlinear system and may have different solutions such as one, two, or no real solutions at all. Solving these systems requires more advanced strategies, such as substitution or elimination, and can involve factoring or using the quadratic formula when dealing with quadratic equations.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). They can have two real solutions, one real solution, or no real solutions when considering the real number system.

In the given system, the equation \(2x^{2} - 4x = 0\) is a quadratic equation. Factoring is often used to solve such equations since it allows for breaking down the equation into two binomials. After factoring, we find the values of \(x\) that make each binomial equal to zero, these values are our solutions to the quadratic equation.

Graphical Solution Verification

Graphical solution verification is a visual method used to confirm the solutions of a system of equations. After solving the system algebraically, you draw the graphs of the given equations on the same coordinate plane.

The points of intersection represent the solutions to the system. For the provided system of equations, plotting the graphs would involve drawing a parabola for \(x^{2} + y = 0\) and a downward-opening parabola for \(x^{2} - 4x - y = 0\). The points where these parabolas intersect are the solutions that you've calculated algebraically. By accurately graphing these functions, you can visually verify that the solutions are correct when the intersections match the solutions found through the substitution method.