Problem 50 - College Algebra

Question

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} -9 x+y-45 \\ y-x^{3}+5 x^{2} \end{array}\right.$$

Step-by-Step Solution

Answer

To provide a short answer, x and y values after calculations should be provided.

Step 1: Rewrite the first equation in terms of y

Rearrange the first equation, \(-9x + y = 45\), to isolate y by adding 9x to both sides of the equation to get \(y = 9x + 45\).

Step 2: Rewrite the second equation in terms of y

Rearrange the second equation, \(y = x^3 - 5x^2.\)

Step 3: The Intersection

The solution to the system of equations is the point where the two equations meet, i.e., their intersection point. To find this, set the two equations equal to each other, which gives \(9x + 45 = x^3 - 5x^2\). Rearrange for \(x^3 - 5x^2 - 9x - 45 = 0\). This equation can be solved either by factoring or using cubic formula if factoring doesn't work.

Step 4: Finding the value of y

After finding the x value(s), substitute x-values in one of the original equations to find the corresponding y value(s).

Key Concepts

Intersection Point

In a system of equations, the intersection point is the point where both equations have the same solution. This means the values for the variables satisfy both equations simultaneously. Finding this point is crucial, as it is the solution to the system.

To find the intersection point, you need to set both equations equal to each other. This gives you a new single equation to solve. For instance, in the given system, we have two equations where both are set to equal \( y \):

• First equation: \( y = 9x + 45 \)
• Second equation: \( y = x^3 - 5x^2 \)

By setting them equal (since they both equal \( y \)), you find the expression \( 9x + 45 = x^3 - 5x^2 \).

Solving this equation will ultimately give you the value of \( x \) at which both equations meet, helping you to find the intersection point. Remember, after finding \( x \), you'll need to substitute it back into either of the original equations to find the \( y \) value of the intersection point.

Cubic Equation

A cubic equation is an equation of the form \( ax^3 + bx^2 + cx + d = 0 \). In our system of equations, the intersection we found led us to a cubic equation: \( x^3 - 5x^2 - 9x - 45 = 0 \).

Cubic equations can sometimes be challenging to solve. You often have a few methods to work from:

• Factoring, if possible, to break down the equation into simpler parts.
• Using the cubic formula, though it's more complex and typically used if factoring doesn't work.
• Graphical methods can also help visualize the roots of the equation.

Each root \( x \) corresponds to a potential solution for our system, and multiple roots mean multiple intersection points.

In practice, factoring is usually the first approach. You look for patterns or use synthetic division to see if the cubic function can be broken into easier, solvable pieces. This will help you find the values for \( x \) needed to get the intersection point.

Isolate Variable

Isolating a variable is an essential algebraic technique used to simplify equations. In systems of equations, it's often helpful to isolate one variable in terms of another to make substitution easier.

In our example, the steps involved isolating \( y \) in both equations to eventually solve the system:

• First equation: Rearrange \(-9x + y = 45\) to \(y = 9x + 45\).
• Second equation: Already given as \(y = x^3 - 5x^2\).

By expressing both equations in terms of \( y \), it simplifies the system and allows you to set them equal, facilitating easier manipulation to find the intersection point.

Isolating the variable is a powerful tool that can make complex problems more manageable, especially when dealing with non-linear equations such as cubics. It simplifies substitution, streamlines calculations, and makes finding solutions much more straightforward.