Problem 38 - Precalculus: Functions and Graphs

Question

Exer \(37-40:\) Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$\left\\{\begin{array}{l} 5 x^{2}+y^{2}=189 \\ 3 x+y=7 \end{array}\right.$$

Step-by-Step Solution

Answer

The points of intersection are (5, -8) and (-2, 13).

Step 1: Express One Variable in Terms of the Other

From the linear equation \(3x + y = 7\), express \(y\) in terms of \(x\):\[y = 7 - 3x\]

Step 2: Substitute Linear Equation into Quadratic Equation

Substitute \(y = 7 - 3x\) into the quadratic equation \(5x^2 + y^2 = 189\):\[5x^2 + (7 - 3x)^2 = 189\]

Step 3: Expand and Simplify the Equation

Expand the equation \((7 - 3x)^2\) and simplify:\[(7 - 3x)^2 = 49 - 42x + 9x^2\]Substitute back:\[5x^2 + 49 - 42x + 9x^2 = 189\]Combine like terms:\[14x^2 - 42x + 49 = 189\]

Step 4: Move All Terms to One Side

Rearrange the equation and set it to zero:\[14x^2 - 42x + 49 - 189 = 0\]\[14x^2 - 42x - 140 = 0\]

Step 5: Divide the Equation to Simplify

Divide the whole equation by 2 to simplify:\[7x^2 - 21x - 70 = 0\]

Step 6: Use the Quadratic Formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) where \(a = 7\), \(b = -21\), and \(c = -70\):\[x = \frac{-(-21) \pm \sqrt{(-21)^2 - 4 \cdot 7 \cdot (-70)}}{2 \cdot 7}\]Calculate the discriminant \((-21)^2 + 4 \times 7 \times 70 = 441 + 1960 = 2401\).\[x = \frac{21 \pm \sqrt{2401}}{14}\]Calculate \(\sqrt{2401} = 49\).\[x = \frac{21 \pm 49}{14}\]

Step 7: Find Values of x

Calculate each value of \(x\):\[x_1 = \frac{21 + 49}{14} = 5\]\[x_2 = \frac{21 - 49}{14} = -2\]

Step 8: Substitute Back to Find y Values

Plug \(x_1 = 5\) and \(x_2 = -2\) back into the linear equation \(y = 7 - 3x\):For \(x_1 = 5\),\[y = 7 - 3 imes 5 = -8\]For \(x_2 = -2\),\[y = 7 - 3 \times (-2) = 13\]

Step 9: Identify Points of Intersection

The points of intersection are \((5, -8)\) and \((-2, 13)\).

Step 10: Sketch the Graphs

Draw the graph of the parabola \(5x^2 + y^2 = 189\), which is an ellipse, and the graph of the line \(3x+y=7\) on the same coordinate plane. The points \((5, -8)\) and \((-2, 13)\) are where they intersect.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means it includes an

• variable squared (e.g., \(x^2\))
• can typically be written in the form \(ax^2 + bx + c = 0\)

In this exercise, the quadratic equation is part of the system of equations involving:

• \(5x^2 + y^2 = 189\)

To solve a quadratic equation, one often uses various strategies, such as factoring, completing the square, or applying the quadratic formula. For more complex equations like this, substitution of known values can simplify the process. Breaking down the equation into simpler parts such as completing the square and making substitutions helps in finding the solutions to complex systems where quadratic equations are involved.

Linear Equation

Linear equations are equations of the first degree, which implies their graphs form straight lines.

• The standard form of a linear equation in two variables is \(ax + by = c\)
• In this exercise, it is \(3x + y = 7\)

Linear equations can be easily solved algebraically for one of the variables. You can discover this through rearrangement, enabling substitution into other equations, just like outlined in the given example with \(y = 7 - 3x\). Understanding how linear equations function within a system allows you to manipulate them effectively, facilitating the solution of the system by substitution or elimination. They are foundational in understanding more complex systems including both linear and non-linear equations.

Intersection Points

The intersection points of two graphs are the values of \(x\) and \(y\) that satisfy both equations simultaneously.

• These points show where the two graphs meet on the graph
• In this case, calculate them by solving the system of equations together

Through a systematic approach—by solving for \(x\) and substituting back to find \(y\)—the intersection points were discovered as

• \((5, -8)\) and \((-2, 13)\)

Once calculated, these points reveal the solution to this real-world problem involving both a linear and a quadratic equation. Comprehending intersection points deeply is crucial, as they represent shared solutions in graphing systems of equations.

Graphing Systems of Equations

Graphing systems of equations provides a visual representation of where two or more equations meet or cross each other. It's a combination of graphical displays illustrating the:

• Quadratic equation as an ellipse in this case
• Linear equation as a straight line

On the coordinate plane, these intersections come to life at the calculated points of

• \((5, -8)\) and \((-2, 13)\)

Sketching out these graphs aids in understanding complex algebraic solutions and brings clarity to the solutions derived from solving the equations algebraically. It’s an essential tool for comprehending multi-equation systems, enhancing the ability to visualize the mathematical concepts involved.