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Exercises Systems of equations

Example 3 from Algebra and Trigonometry, 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables

Solving a System of Nonlinear Equations Representing a Circle and an Ellipse

Solve the system of nonlinear equations.

\begin{array}{r} x^{2} + y^{2} = 26 & (1) \\ 3 x^{2} + 25 y^{2} = 100 & (2) \end{array}

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Let’s begin by multiplying equation (1) by $−3,$ and adding it to equation (2).
$\frac{\begin{array}{l} \begin{array}{l} (- 3) (x^{2} + y^{2}) = (- 3) (26) \\ - 3 x^{2} - 3 y^{2} = - 78 \end{array} \\ 3 x^{2} + 25 y^{2} = 100 \end{array}}{22 y^{2} = 22}$
After we add the two equations together, we solve for $y .$

$\begin{array}{l} y^{2} = 1 \\ y = \pm \sqrt{1} = \pm 1 \end{array}$
Substitute $y = \pm 1$ into one of the equations and solve for $x .$

$\begin{array}{l} x^{2} + {(1)}^{2} = 26 \\ x^{2} + 1 = 26 \\ x^{2} = 25 \\ x = \pm \sqrt{25} = \pm 5 \\ x^{2} + {(−1)}^{2} = 26 \\ x^{2} + 1 = 26 \\ x^{2} = 25 = \pm 5 \end{array}$
There are four solutions: $(5, 1), (−5, 1), (5, −1), and (−5, −1) .$ See Figure 7.

Figure 7

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the solution set for the given system of nonlinear equations.

\begin{matrix} 4 x^{2} + y^{2} & = & 13 \\ x^{2} + y^{2} & = & 10 \end{matrix}

How did it go?

PreviousFinding the Intersection of a Circle and a Line by Substitution Next Graphing an Inequality for a Parabola

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 11.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.