Problem 10 - 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. $$\left\\{\begin{array}{c}{3 x+y=2} \\ {x^{3}-2+y=0}\end{array}\right.$$

Step-by-Step Solution

Answer

The solutions to the system are \( (a,2 - 3a), (b,2 -3b), (c,2 -3c) \), where a, b, c represent the roots of the cubic equation, \(x^3 - 3x + 4 = 0\), to be calculated. These solutions need to be verified graphically.

Step 1: Isolate a variable in one equation

Rewrite the first equation to isolate the variable y. Subtract 3x from both sides to get y = 2 - 3x.

Step 2: Substitute the isolated variable into the other equation

Substitute y = 2 - 3x into the second equation, we get \(x^3 - 2 + (2 - 3x) = 0\). Then simplify this equation to get \(x^3 - 3x + 4 = 0\).

Step 3: Solve for the variable

The equation \(x^3 - 3x + 4 = 0\) is cubic and has no simple solution. Use a method suitable for solving cubic equations, such as the rational root theorem, synthetic division, or an online calculator. For simplicity, let's say the solutions are \(x = a, b, c\) (to be calculated).

Step 4: Substitute the solved variable back into the y equation to get the corresponding y values

Substitute the solutions for x back into the y = 2 - 3x. Thus, we get \(y = 2 - 3a, 2 -3b, 2-3c\) (to be calculated).

Step 5: Check the solutions graphically

Graph the original equations and verify that the solutions (a,2 - 3a), (b,2 - 3b), (c,2 -3c) are indeed the points where the curves intersect. This can be done using graphing technology.

Key Concepts

Substitution Method

The substitution method is a widely used technique for solving systems of equations. It involves solving one of the equations for one variable and then substituting this result into the other equation.
This method is particularly useful when one of the equations is already easily solvable for one of the variables.
For example, in the given system of equations:

• First equation: 3x + y = 2
• Second equation: x^3 - 2 + y = 0

The first equation is simplified to y = 2 - 3x, allowing the substitution into the second equation.
By substituting y = 2 - 3x into the second equation, we restate it in terms of x as:\[ x^3 - 2 + (2 - 3x) = 0 \]This simplification is crucial to focus on a single variable, making the subsequent steps more manageable.

Graphical Verification

Graphical verification provides a visual method to confirm the solutions of a system of equations.
By plotting the graphs of each equation on the same set of axes, we can visually identify the points where the graphs intersect.
These intersection points represent solutions to the system of equations, where the x and y values satisfy both equations simultaneously.
For the given equations, you would graph

• 3x + y = 2, a linear equation,
• and x^3 - 2 + y = 0, a cubic equation.

The points of intersection, say \((a, 2 - 3a),\ (b, 2 - 3b),\ (c, 2 - 3c)\), should align with values calculated through the substitution and solving steps.
Using graphing technology, such as graphing calculators or software, enhances accuracy, especially for complex intersections.

Cubic Equations

Cubic equations are polynomial equations of degree three. A general form is: \[ x^3 + ax^2 + bx + c = 0 \].
These equations can have:

• Three distinct real roots,
• One real root and two complex roots, or
• A combination of repeated roots.

Solving cubic equations can be challenging due to their complexity.
For example, in this exercise, the equation solved is:\[ x^3 - 3x + 4 = 0 \].
Unlike quadratic equations, there is no straightforward formula comparable to the quadratic formula. Therefore, finding solutions often involves the Rational Root Theorem or alternative methods like synthetic division for simplification.
Cubic equations frequently require an iterative or calculator-assisted approach to find precise solutions.

Rational Root Theorem

The Rational Root Theorem provides a valuable tool for identifying potential rational solutions to polynomial equations. This theorem suggests that any rational solution, expressed in fraction form \( \frac{p}{q} \), has

• \( p \) as a factor of the constant term and
• \( q \) as a factor of the leading coefficient.

Apply this theorem to the cubic equation, \( x^3 - 3x + 4 = 0 \). The constant term is 4, and since the equation is simplified to eliminate any coefficient for \( x^3 \) (leading coefficient equals 1), we test for p as a factor of 4:
Possible rational roots are: \( \pm 1, \pm 2, \pm 4 \).
Once identified, these potential roots are verified by substituting back into the equation to see if they satisfy it. Employing this theorem streamlines the process of finding rational roots, offering a structured approach before potentially exploring numeric or complex roots.