Problem 40 - Algebra and Trigonometry

Question

Solve each system by the method of your choice. $$ \left\\{\begin{array}{l} {(x-1)^{2}+(y+1)^{2}=5} \\ {2 x-y=3} \end{array}\right. $$

Step-by-Step Solution

Answer

The solutions to the system of equations are (0,-5) and (2,1)

Step 1: Rearrange the linear equation

Rearrange the linear equation \(2x - y = 3\) to make y the subject. This will give \(y = 2x - 3\)

Step 2: Substitute y into the circle equation

Substitute \(y = 2x - 3\) into the circle equation. This will give the equation \((x-1)^2 + (2x-3+1)^2=5 \) which simplifies to \((x-1)^2 + (2x-2)^2=5\) which also simplifies to \(x^2-2x+1 + 4x^2-8x+4=5\)

Step 3: Solve the Equation

Simplify equation to \(5x^2-10x=0\). Factoring out gives \(5x(x - 2)= 0\). Setting this equal to zero gives the roots x = 0 and x = 2.

Step 4: Solve for Corresponding y Values

Substitute x=0 and x=2 into equation for y to get corresponding y values. When x=0, y=2(-1)-3=-2-3=-5. When x=2, y=2(2)-3=4-3=1.

Key Concepts

Understanding Linear Equations

Linear equations are a fundamental concept in algebra. They typically appear in the form of y = mx + c, where m is the slope of the line, and c is the y-intercept. These equations represent straight lines in a two-dimensional plane.

• Slope (m): The slope indicates the steepness and direction of the line. A positive slope means the line goes upwards, while a negative slope means it goes downwards.
• Y-intercept (c): The y-intercept is the point where the line crosses the y-axis. It is the value of y when x is zero.

In our specific problem, the linear equation is 2x - y = 3. To solve the system, we first rearrange this equation to make y the subject:
\[ y = 2x - 3 \]
This makes it easier for us to perform substitution later on. By isolating y, we express it in terms of x, preparing to use it in conjunction with the circle equation.

Understanding Circle Equations

A circle equation is another key concept in coordinate geometry. It describes all the points that form a circle on a plane. The most familiar form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\). Here,

• \((h, k)\) is the center of the circle.
• \(r\) is the radius.

For our exercise, the circle equation is \((x - 1)^2 + (y + 1)^2 = 5\).

• The center of this circle is at \((1, -1)\) and the radius is \(\sqrt{5}\).

This equation means all points \((x, y)\) that satisfy it will lie on the circumference of the circle with these particular center coordinates and radius.

The Substitution Method Explained

The substitution method is a powerful technique for solving systems of equations, especially when dealing with a linear and non-linear equation, such as a circle equation.
Here's how the substitution method works:

• Step 1: Solve one of the equations for one of the variables. In our exercise, we solved the linear equation for y, obtaining \(y = 2x - 3\).
• Step 2: Substitute this expression for y into the second equation, the circle equation \((x - 1)^2 + (y + 1)^2 = 5\). This gives us an equation solely in terms of x, making it easier to solve for x.

By substituting \(y = 2x - 3\) into the circle equation, \((2x - 3 + 1)^2\) becomes part of the equation. Simplifying this helps to eventually determine the x-values by solving the resulting quadratic equation.
The power of substitution lies in reducing the complexity to a simpler, one-variable equation, allowing us to solve systems of equations systematically and efficiently.