Problem 70 - A Graphical Approach to College Algebra

Question

Solve each nonlinear system of equations analytically. $$\begin{array}{r}x^{2}+y^{2}=5 \\\\-3 x+4 y=2\end{array}$$

Step-by-Step Solution

Answer

The solutions are \((1.52, 1.64)\) and \((-2, -1)\).

Step 1: Identify the System of Equations

Let's start by identifying the system of equations that we need to solve: 1. The first equation is a circle given by \( x^2 + y^2 = 5 \).2. The second equation is a line described by \( -3x + 4y = 2 \). Our goal is to find the points \((x, y)\) that satisfy both equations simultaneously.

Step 2: Solve the Linear Equation for y

To make solving easier, start by solving the linear equation for \( y \). From the equation \(-3x + 4y = 2\), we can express \(y\) in terms of \(x\):\[ 4y = 3x + 2 \]\[ y = \frac{3x + 2}{4} \].

Step 3: Substitute y in the Circle Equation

Substitute \( y = \frac{3x+2}{4} \) into the circle's equation \( x^2 + y^2 = 5 \):\[ x^2 + \left(\frac{3x + 2}{4}\right)^2 = 5 \].

Step 4: Simplify the Substituted Equation

Simplify the expression - expand and re-arrange:\[ x^2 + \frac{(3x + 2)^2}{16} = 5 \]\[ x^2 + \frac{9x^2 + 12x + 4}{16} = 5 \]Multiply through by 16 to clear the fraction:\[ 16x^2 + 9x^2 + 12x + 4 = 80 \]Combine like terms:\[ 25x^2 + 12x + 4 - 80 = 0 \]\[ 25x^2 + 12x - 76 = 0 \].

Step 5: Solve the Quadratic Equation

Solve \(25x^2 + 12x - 76 = 0\) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 25 \), \( b = 12 \), and \( c = -76 \).Calculate the discriminant:\[ b^2 - 4ac = 12^2 - 4 \times 25 \times (-76) = 144 + 7600 = 7744 \]Solve for \( x \):\[ x = \frac{-12 \pm \sqrt{7744}}{50} \]\( \sqrt{7744} = 88 \), therefore:\[ x_1 = \frac{-12 + 88}{50} = \frac{76}{50} = 1.52 \]\[ x_2 = \frac{-12 - 88}{50} = \frac{-100}{50} = -2 \].

Step 6: Substitute x back to Find y

Substitute \(x_1 = 1.52\) into \(y = \frac{3x + 2}{4}\):\[ y = \frac{3(1.52) + 2}{4} = \frac{4.56 + 2}{4} = 1.64 \]Substitute \(x_2 = -2\):\[ y = \frac{3(-2) + 2}{4} = \frac{-6 + 2}{4} = -1 \].

Step 7: Write the Solutions

The solutions to the system are the ordered pairs found by the above computations: \( (x, y) = (1.52, 1.64) \) and \( (-2, -1) \).

Key Concepts

Analytical Solutions

An analytical solution provides an exact expression or set of expressions that describe the solution to a problem. This method is fundamental in mathematics because it provides clarity and an exact understanding of how the values relate to one another. When tackling nonlinear systems of equations, like circles and lines, identifying a solution analytically means finding precise values for unknown variables rather than approximations.
To solve such systems:

• First, identify the equations involved and how they interact with each other.
• Next, manipulate one equation to isolate a variable and express it in terms of other variables.
• Finally, substitute this expression back into the other equation to find the exact values of all unknowns.

This step-by-step process not only reveals the solution but also deepens understanding of the relationships governing the system.

Circle Equation

A circle's equation describes all the points that are equidistant from a central point. In this exercise, the circle equation is given by \(x^2 + y^2 = 5\). Here:

• The term \(x^2 + y^2\) represents the square of the distance from the origin (0,0) in a Cartesian plane.
• The number 5 is the square of the circle's radius.

So, we are looking for all points \( (x, y) \) that sit on the boundary of a circle centered at the origin with a radius of approximately 2.236 (since \( \sqrt{5} \approx 2.236 \)).
This is critical, as any solution to our equations must satisfy this geometric constraint.

Line Equation

A line equation represents all points along a straight path in a plane. Given by \(-3x + 4y = 2\), this equation can be transformed into different forms to make calculation more convenient. By rearranging, we solve it for \(y\):

• The equation becomes \(4y = 3x + 2\), and dividing by 4 gives \(y = \frac{3x + 2}{4}\).

This simplified form allows for easier substitution into the circle equation. By finding the expression for \(y\), we create a bridge between the two original equations, facilitating a comprehensive solution to the system.

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions of quadratic equations, those in the form \(ax^2 + bx + c = 0\). In this case, after substituting the expression for \(y\) into the circle equation and simplifying, we arrive at the quadratic equation \(25x^2 + 12x - 76 = 0\).
To solve it, the quadratic formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

where \(a = 25\), \(b = 12\), and \(c = -76\). Calculating the discriminant \(b^2 - 4ac\) and its square root gives the essential component for solving for \(x\).
Finally, substituting these solutions back into the linear equation finds the corresponding \(y\) values, yielding the complete solution set for the system of equations.