Problem 42 - Algebra and Trigonometry

Question

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{aligned} x^{2}+y^{2} &=17 \\ x^{2}-2 x+y^{2} &=13 \end{aligned}\right.\)

Step-by-Step Solution

Answer

The solutions are \( (2, 3.61) \) and \( (2, -3.61) \).

Step 1: Rewrite the Equations

Start by rewriting the given equations for clarity. We have two equations: 1. \( x^2 + y^2 = 17 \) 2. \( x^2 - 2x + y^2 = 13 \)

Step 2: Simplify Second Equation

Subtract the second equation from the first to eliminate \( y^2 \): \( (x^2 + y^2) - (x^2 - 2x + y^2) = 17 - 13 \). This simplifies to \( 2x = 4 \), so \( x = 2 \).

Step 3: Solve for y

Substitute \( x = 2 \) back into the first equation: \( 2^2 + y^2 = 17 \). This simplifies to \( 4 + y^2 = 17 \) or \( y^2 = 13 \). Therefore, \( y = \pm \sqrt{13} \).

Step 4: Identify Solution Pairs

We have \( y = \pm \sqrt{13} \). Thus, the solutions are: \( (x, y) = (2, \sqrt{13}) \) and \( (2, -\sqrt{13}) \).

Step 5: Convert to Decimal Form

Calculate the decimal form of \( \sqrt{13} \): \( \sqrt{13} \approx 3.61 \). Therefore, the solutions are \( (2, 3.61) \) and \( (2, -3.61) \).

Key Concepts

System of Equations

When dealing with a system of equations, we are working with multiple equations that share common variables. In this case, the variables are \( x \) and \( y \). Our goal is to find values for these variables that satisfy both equations at the same time. It's like searching for points where two curves intersect on a graph. The given equations represent two circles:

• \( x^2 + y^2 = 17 \) is a circle centered at the origin with a radius of \( \sqrt{17} \)
• \( x^2 - 2x + y^2 = 13 \) can be rewritten by completing the square to represent another circle.

Visualizing these circles and their intersection points helps us understand where the solutions might lie.
This is the essence of the graphical method: finding the shared solutions visually by drawing the equations on the same plane.

Solution Pairs

Solution pairs are the values of \( x \) and \( y \) that solve a system of equations simultaneously. For this problem, it means identifying the points where the two circles intersect.After simplifying our equations, we found the values of \( x \) and corresponding \( y \) values that make both equations true:

• First, we simplified using subtraction to find \( x = 2 \)
• Substituted \( x = 2 \) back into the original equations to solve for \( y \)
• Calculated that \( y = \pm \sqrt{13} \), giving us two pairs: \((2, \sqrt{13})\) and \((2, -\sqrt{13})\)

These pairs represent the points of intersection of the circles, showing us where both equations hold true.

Decimal Approximation

Decimal approximation helps express solution pairs in more practical, numerical terms. While \( \pm \sqrt{13} \) is a precise answer, everyday calculations prefer decimals. To convert \( \sqrt{13} \) to decimal form, we determine that \( \sqrt{13} \approx 3.61 \). This helps us express our solution pairs in familiar terms:

• \( (2, 3.61) \) and \( (2, -3.61) \)

This approximation is crucial in applications where exact values are cumbersome or unnecessary. Remember, always round your final result to the required precision, in this case, two decimal places.

Simplification of Equations

Simplification reduces the complexity of equations, making them easier to solve. In this problem, simplification was key to finding the values of \( x \) and \( y \). Here's how we simplified:

• By subtracting the second equation from the first, we eliminated \( y^2 \), simplifying our system to \( 2x = 4 \)
• We solved for \( x \) quickly, getting \( x = 2 \)

This step transformed the system into a more manageable format, enabling us to solve for \( y \) using basic algebra. Simplifying equations can involve removing terms, combining like terms, or factoring, and is an essential skill for efficiently solving mathematical systems.