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Chapter 13: Problem 50

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

Short Answer

Expert verified
The solutions to the system of equations are (-5,0), (-3,18), and (3,72).

Step by step solution

01

Express One Variable in terms of Other

The first step in this procedure is to isolate one of the variables in one of the equations. The linear equation \( -9x + y = 45 \) is best for this. Re-arranging this equation to express \(y\) in terms of \(x\) will result in \( y = 9x + 45\).
02

Substitute the Isolated Variable Into the Second Equation

The next step is to substitute \(y\) from our derived equation into the second given equation to form a cubic equation in one variable (in this case, \(x\)). The equation after substitution becomes: \(9x + 45 = x^3 + 5x^2 \).
03

Solve for \(x\)

The equation \(9x + 45 = x^3 + 5x^2 \) simplifies to \( x^3 + 5x^2 - 9x - 45 = 0 \). Solving this equation yields \(x = -5, x = -3, x = 3\).
04

Substitute \(x\) back into y = 9x + 45 and Solve for \(y\)

Substitute \(x\) Back Into the first equation and Solve for \( y \). When \( x = -5 \), \( y = 9*(-5) + 45 = 0 \). When \( x = -3 \), \( y = 9*(-3) + 45 = 18 \). When \( x = 3 \), \( y = 9*3 + 45 = 72 \). These are the corresponding values of \( y \). Therefore, the solution pairs are (-5,0), (-3,18), and (3,72).

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Most popular questions from this chapter

(GRAPH CANT COPY) Find the coordinates of the vertex for the horizontal parabola defined by the given equation. $$x=y^{2}-6 y+6$$

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